In mathematics, in the realm of group theory, a group is said to be **superperfect** when its first two homology groups are trivial: *H*_{1}(*G*, **Z**) = *H*_{2}(*G*, **Z**) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. 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.

A finite perfect group is superperfect if and only if it is its own universal central extension (UCE), as the second homology group of a perfect group parametrizes central extensions.

For example, if *G* is the fundamental group of a homology sphere, then *G* is superperfect. The smallest finite, non-trivial superperfect group is the binary icosahedral group (the fundamental group of the PoincarĂ© homology sphere).

The alternating group *A*_{5} is perfect but not superperfect: it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE, and is superperfect). More generally, the projective special linear groups PSL(*n*, *q*) are simple (hence perfect) except for PSL(2, 2) and PSL(2, 3), but not superperfect, with the special linear groups SL(*n*,*q*) as central extensions. This family includes the binary icosahedral group (thought of as SL(2, 5)) as UCE of *A*_{5} (thought of as PSL(2, 5)).

Every acyclic group is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.