Computing group cohomology rings from the Lyndon–Hochschild–Serre spectral sequence

We describe a method for computing presentations of cohomology rings of small finite p-groups. The description differs from other accounts in the literature in two main respects. First, we suggest some techniques for improving the efficiency of the obvious linear algebra approach to computing projective resolutions over a group algebra. Second, we use an implementation of the multiplicative structure of the Lyndon–Hochschild–Serre spectral sequence for determining how much of a projective resolution needs to be computed in order to obtain a presentation of the cohomology ring.