John Conway <conway@math.princeton.edu> writes:
Since the two best examples are A(4) = 2^2 : 3 and 2^4 : 3 it would seem that 2^6 : 3 deserves examination. Let me try to do that now (despite the fact that I have to lecture in a few minutes).
... I presume 7/32 beats 5/24, but don't have time to check.
Well, of course it does. It seems that 2^6:3 is SmallGroup(192,1541), and yes, it has a s.s=n density of 5/32. I haven't got a way of forming the semidirect product os something that large yet. And pretty much by luck, scanning the 1090235 groups of order 768 from the highest numbered down, the first one that had a derived subgroup of 2^8 was SmallGroup(768,1085321), and it's got a s.s=n density of 85/384 . Assuming that group is 2^8:3, we've got density of 2(1-2^-k)/9 in 2^k:3 for 0(?),2,4,6,8 . I didn't find anything over 1/6 in groups of size 80 or 448. I couldn't check 320. Dan