An acyclic group is a group "G" whose classifying space "BG" is acyclic; in other words, all its (reduced) homology groups vanish, i.e., formula_3, for all formula_4. Every acyclic group is thus a perfect group, meaning its first homology group vanishes: formula_5, and in fact, a superperfect group, meaning the first two homology groups vanish: formula_6. The converse is not true: the binary icosahedral group is superperfect (hence perfect) but not acyclic.