Somehow this question made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S_5,A_6,S_6,A_7,A_8,S_8$ are the only instances of symmetric/alternating groups that are not quotients of $Z/2Z*Z/3Z=PSL_2(Z)$ (see this MathSciNet entry, which I just found). Does anyone have an idea of a conceptual explanation for this fact?
Edit: I also find this article which mentions the same result. It's quite interesting that the positive part (for all $n>8$ these groups are quotients) is proved using Bertrand's postulate. I think it's cool that Bertrand's postulate can be used for group theory.