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: $A_5)$ A_5$) is a counterexample. For $N=2$, the binary icosahedral group $SL_2(\mathbb{F}_5)$ 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 $M23$. 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.
Has any progress been made on this, or
Is there another known group with a related questionlarger $N\ge 5$ before homology becomes nontrivial?In particular, are
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.
