Is this normal for a programming team of this size?
My instinct is "maybe", depending on technique.
With 6 people in the team, and by definition at least 2 people must be in a conversation, there must be people who are in conversation at least 1/3 of the time in order for the talk to be "non-stop".
That seems a bit high to me if they program alone, given that presumably they are not carefully scheduling their conversations to cover the available time. Therefore some people must be engaged in conversation considerably more than 1/3 the time.
However, if they are pair-programming, which is not uncommon, then the proportion of time they spend in conversation is much greater than if they are not. You should find out whether or not they are -- if they are not then I would say they're talking at their desks an unusual amount. They could probably stand to talk less or to take some of it elsewhere. If they are pair-programming then they probably should not be in the same room as you for the same reason that people whose job it is to make phone calls all day should not be in the same room as you.
Unfortunately, employers for various reasons favour open-plan offices even though they are demonstrably less productive environments for many tasks[*]. Management has to trade off the cost of office space, the benefits of free communication across the whole employee base as opposed to clustering information by room, and direct productivity per person. They might prefer to pay for 50% more employees who are only 67% as productive, and that is their prerogative. If your employer cannot (or chooses not to) provide a quiet place for you, then part of your job is to deal with your poor working environment. At least you're indoors ;-)
The usual response is to listen to music on headphones -- this is still a significant distraction for some people, but anecdotally I think people find that noise they control is far less harmful to productivity and happiness than noise they don't.
[*] Rather, noise demonstrably reduces productivity. My employer has an open plan office, that I don't work in, that is eerily quiet compared with what I've experienced elsewhere. On the occasions I've worked there for a day I haven't found noise to be a problem, since there's a strong norm of keeping conversations quiet and taking large conversations outside. But you can't put that norm in place by yourself.
How can we convince them to quiet down and get back to work or take
the conversation to one of our many conference rooms?
It's likely to be difficult because they have an established work environment and you want to make a large change to it.
Firstly you need to address your feelings about it (this is the paragraph where I get a downvote from the questioner ;-)). You don't like it and it harms your work. However, there is no point characterising it as "abnormal". There is no point thinking of them stopping talking as "getting back to work", since much of the conversation is work. It's not really your business to judge whether they're slacking even if they are. Your colleagues will not respond well to hostility, and regardless of how an office would ideally be, you have no moral high horse here. I happen to agree with you that noisy work environments are a terrible thing to inflict on people, but even so you should think of this as requesting a favour, not enforcing better behaviour or shutting them up.
Then you need to tell them its an issue, they surely won't change their behaviour other than by knowing it's causing problems for others. But you're correct that an email from on high saying that "someone has complained about you" is a terrible way to do that. It sounds as though their manager has no sympathy at all for your request. This is an issue to deal with in itself. In any company, but especially in an open-plan office, there needs to be a way to consider how teams' behaviour affects other teams. Find it, or create it ad-hoc by talking to people you think will be more sympathetic.
If they're open to change once they learn that others would benefit from quiet, then they'll probably accept a certain amount of nagging on the subject. You could perhaps agree with them that they will try to take long conversations out of the room, and that you are permitted to tell them that their conversation is running long without them feeling harassed. The only way to establish this relationship for sure is to make it explicit. Ideally there should be some reciprocity: can they find anything in your team's behaviour that could improve things for others?
If they are pair programming then basically you can't do anything about it other than by ending open-plan. They must do this at their normal desk by definition of "normal desk", and they must talk most of the time they're doing it.
An aside on open-plan offices: the concept was driven forward in the 50s and 60s as a great leveller, putting everyone (or at any rate anyone working on that floor) in one shared space. Architecturally speaking this was exciting stuff, using a building to improve relationships and literally remove status barriers that impeded the work of the company. The benefits of this may well have outweighed the costs in individual productivity on "head-down" focussed work, although the latter needs to be considered.
Call me a cynical curmudgeon, but some companies have used "open plan" as a way of sticking rank-and-file workers into the equivalent of a pre-50s typing pool plus potplants, while anybody of any consequence has a private office "for reasons of confidentiality". They may still claim the same benefits in terms of communication when actually they're just saving office footprint. If so, avoid. One kind of person needs permanent access to a private office and that isn't even C-level executives, it's HR (and possibly finance and legal, but the point is the nature of the work not the seniority).
As a consequence of the effect of noise, there's a strong instinct that it's the responsibility of any worker in an open-plan office to be as quiet as possible at all times. Certainly nobody benefits from distractions in themselves, but it completely misses the point to expect everyone to work as if under a vow of silence :-)