I stumbled across this question in a seminar-paper a long time ago:

*Does there exist a positive integer* $N$ *such that if* $G$ *is a finite group with* $\bigoplus_{i=1}^NH_i(G)=0$ *then* $G=\lbrace 1\rbrace$?

I believe this to still be an open problem. For $N=1$, any perfect group (ex: $A_5$) is a counterexample. For $N=2$, the binary icosahedral group $SL_2(F_5)$ suffices (perfect group with periodic Tate cohomology). And I found in one of Milgram's papers a result for $N=5$, the sporadic Mathieu group $M_{23}$. Note that this question is answered for infinite groups, because we can always construct a topological space (hence a $BG$ for some discrete group $G$) with prescribed homologies.

**Is there another known group with a larger $N\ge 5$ before homology becomes nontrivial?
Are there any classifications of obstructions in higher homology groups?**

[[Edit]]: Another view. A group is $\textit{acyclic}$ if it has trivial integral homology. There are no nontrivial finite acyclic groups. Indeed, a result of Richard Swan says that a group with $p$-torsion has nontrivial mod-$p$ cohomology in infinitely many dimensions, hence nontrivial integral homology.

`$H^*(G)\to H^*(<g>)$`

is nontrivial. I very much doubt that there is a uniform $n$ so that if a finite group has vanishing cohomology in degrees less than $n$, that it vanishes, but I can't give you any examples of groups with large vanishing ranges. $\endgroup$ – Ben Wieland Jan 26 '11 at 5:43