 In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.The first homology group of a group is the abelianization of the group itself, since the homology of a group "G" is the homology of any Eilenberg–MacLane space of type "K"("G", 1); the fundamental group of a "K"("G", 1) is "G", and the first homology of "K"("G", 1) is then abelianization of its fundamental group.Thus, if a group is superperfect, then it is perfect.