 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 Poincaré homology sphere).