@Brokenbiscuit you have hit the nail on the head. My comments about fitting in are not to change my daughter to be something she is not, but to help her find her place within her existing peer group.
FWIW, I think you're absolutely right Zodle. I was an academic outlier as a child, but I was also very shy. I spent a lot of time reading, and was encouraged to pursue academic interests. Socially, I spent most of my time with one other child - also very academic. My parents weren't at all pushy, but encouraged us to learn and follow our natural interests. Unfortunately, due to my natural shyness and my dependence on this one very academic friend, I didn't get much practice at how to relate to other kids my age. I didn't develop those skills, and when my friend and I went to separate secondary schools, I found that I simply didn't know how to build relationships with others. I did learn eventually, but I still am left with the hangovers from that time and I am very socially anxious as a result. I really wish my parents had invested a little more effort in getting me to mix with a range of "ordinary" kids. I didn't need help with learning or developing my thinking skills - I was brilliant at those things already. What I really needed help with was learning how to chat to other kids my age, how to relate to people of all different abilities and how to build friendships with the people around me.
My dd is very different to me. She instinctively knows how to talk to anyone. She does have quite a few older friends, but she also has a few younger ones as well as lots among her own age group. And she sees some adults as her friends too! Her closest friends at school do tend to be very able academically, but she is also very friendly with some others who are not at all academic - they may not be "clever" but she is thankfully able to appreciate their other good qualities.
OP, the children at the group I went to weren't rude as such. Just lacking in self-awareness and basic turn-taking in conversations etc. The parents seemed to have placed a disproportionate emphasis on academic intelligence, over and above other aspects of their development. I suspect that the assumption that any social difficulties were linked to the giftedness often got in the way of the parents actually doing anything to address those difficulties - social ineptitude was dismissed as an inevitable consequence of giftedness rather than as a problem that needed to be addressed. I felt sorry for the kids.
OP, there are indeed a lot of gifted groups in the UK and in other countries, but I suspect that most are similar to the one I attended. The thing about truly gifted children is that they are already different from their peers. These gifted groups just underline and emphasise those differences, and I'm not at all sure that's helpful. I believe that sensible parents don't dwell on those differences (and therefore on the giftedness) but rather encourage their children to see the many, many things that they share in common with other kids, regardless of academic ability. If we teach them how to relate confidently and effectively to people of all abilities, that's likely to have a profound impact on their long-term success and happiness. Their intelligence will not disappear just because we fail to focus on it.
I agree with those who advise against special gifted groups and recommend other activities that enable their children to mix with a wide range of other kids.