Finite class 2 nilpotent and Heisenberg groups

Abstract We present a structural description of finite nilpotent groups of class at most 2 using a specified number of subdirect and central products of such groups that are 2-generated. As a corollary, we show that any such group is isomorphic to a subgroup of a Heisenberg group satisfying certain properties. The motivation for these results is of a topological nature as they can be used to give lower bounds to the nilpotently Jordan property of the birational automorphism group of varieties and the homeomorphism group of compact manifolds.