Computing p-groups with trivial Schur multiplicator

Recently the author has shown that there are only finitely many p-groups with trivial Schur multiplicator of a given coclass if p is an odd prime. This paper introduces an algorithm to compute these finitely many p-groups for any given prime and coclass. As an application, we classify these groups for coclass 1, give a conjectural description for them for coclass 2 and determine them for coclass 3 and p∈{3,5,7} and coclass 4 and p=3. Based on the experimental evidence, we conjecture that the order of a p-group with trivial Schur multiplicator and coclass r is bounded by a function b(r) which is independent of the prime p.