Cohomology of uniserial p-adic space groups with cyclic point group

Let p be a fixed prime number and let R denote a uniserial p-adic space group of dimension dx=px−1(p−1) and with cyclic point group of order px. In this short note we prove that all the quotients of R of size bigger than or equal to pdx+x have isomorphic mod p cohomology groups. In particular, we show that the cohomology groups of sufficiently large quotients of the unique maximal class pro-p group are isomorphic as Fp-modules.