Why it is different if a child's 'thing' is Maths or writing stories from if it is Music or Dance or Football?
Here the question was not about "My kids really interested in Maths, where can I find some groups where they can enjoy it with peers?" As that would get the same response as music, dance or football I'm sure.
This question was about creating a group where the only common denominator was "gifted and talented", based on the premise that being gifted and talented alone needed different socialisation. Your analogy breaks down because the groups getting football or music advice are not being told they need other talented footballers or musicians to do non-football events. That's the difference.
The advice people have given here is sensible - pick the domain first, and then find groups. It's quite silly to attempt to get a gifted writer, a gifted mathematician, a gifted juggler, a gifted musician, a gifted historian together and expect it to be a success, or any different to a group who don't have various gifts.
It's strange for the OP to quote a study of what happens who people who assess in SAT's as top 1% at 12 solely in maths, as being particularly relevant to a 5-7 year old socialisation. Like most things (such as why top sportsmen take up their sport late) early specialisation often doesn't have brilliant outcomes.
As others have said, socialisation with age appropriate kids happens outside the gifted area, socialisation with age appropriate similarly gifted kids happens in groups specific to their interest, for many this will be impossible as there won't be any similarly gifted and interested groups around. Mixing kids with weak social skills (as almost all 5-7year olds are, and certainly those who call their friends boring, rather than finding how to enjoy the company of their friends in a common activity!) who happen to have wildly different gifts is unlikely to be any more productive at improving social skills than those activities which are away from the talents.
I understand why the OP is wary about all the interactions being with older kids and adults, or in activities where the peers are weaker, being permanently babied or always being teacher to others is not a good environment. I have never heard of a "Renaissance man" at 7, they simply haven't had the time to learn enough, so there will be areas where they are merely average.
Personally I see little wrong with a 2-3 year age gap if the child is socially confident and can demand not to be babied in the areas where the child is gifted, many countries end up with this via advancement in school years, and rarely is it a social issue about the academic areas. (I would be violently against it for the transition periods, puberty, adulthood etc.)
And those talented musicians and footballers will be practicing with peers exactly that sort of age group older.
As I've said before, my personal vision is to encourage development of the non-gifted and talented areas of a child, they'll happen easily and need little encouragement, encourage the others, and also avoid early specialisation.