On the cohomological triviality of the center of the Frattini subgroup

Abstract

We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order $256$. We also show that nonabelian metacyclic $p$-groups, nonabelian groups of maximal nilpotency class and $2$-groups of coclass two satisfy the conjecture.