I'm sure you don't want to hear this but I'll say this anyway.
Mumsnet is full of parents who have " always felt that they would send their child to x type of school " , when they have taken little or no regard of their child's abilities and interests . There are posts on here from parents of babies ans toddlers asking how they can get them into Oxbridge !
Some of them are more interested education as a status symbol or a fashion statement . Others need their children to achieve certain things that they failed to do themselves . While still more want their children to keep up a family tradition , even though it might be entirely the wrong school for their child.
They push and harass and tutor their kids to within an inch of their lives to get them to " fight above their weight " and many of them end up unhappy or struggling in the wrong school / university.
I'm sure you are NOT one of these parents . So I'd suggest that you take a little time to work out what is going on with your DD, who is still very young.
Why do you think she is bright ? Why do you think she is not working to her potential ? Is she falling behind her class ? What does her teacher say? Do you have reason to think she has any specific learning difficulties ? Is she happy at school ? Does she like her teacher and have friends ?
A bright six year old in a good school with a happy home life should not be falling badly behind just because she's in a class of 31. There may well be something else going on. It may just be < whispers > that you child is average and not particularly bright.
I'm not saying don't move her. But please think hard about her needs and not just your own ambitions for her.