On the cohomology of central Frattini extensions

In this paper we provide calculations for the mod p cohomology of certain p-groups, using topological methods. More precisely, we look at p-groups G defined as central extensions 1→V→G→W→1 of elementary abelian groups such that image and the defining k-invariants span the entire image of the Bockstein. We show that if p>dim V−dim W+1, then the mod p cohomology of G can be explicitly computed as an algebra of the form image where image is a polynomial ring on two-dimensional generators and A is the cohomology of a compact manifold which in turn can be computed as the homology of a Koszul complex. As an application we provide a complete determination of the mod p cohomology of the universal central extension image provided image, where n=dim W.