As someone who was privately educated (good selective day school) and can in no way afford it for my own children, I'd strike a note of caution.
Yes, I have good grades, went to a good university (red brick, not top top), got a profession (now left it) etc. However so do loads of people from state schools.
However, as a PP said, I do have an inferiority complex to an extent because although I am clever, I am not super bright like many were and always being in bottom sets, even though I imagine these would have been close to the top in a state school, has had a lasting affect, 20 years on.
Our teachers weren't IMO actually all fantastic (some were) but were able to teach because of the demographic I.e. We were there to learn and put lots of self study in. I also had some rubbish teachers who just couldn't teach to "lower" ability children.
I also missed out on seeing boys as normal creatures and not these exotic species which I think is important growing up.
In summary, you don't know what your daughter will be like later on but don't automatically assume private is best.