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We improve the existing lower bounds on the order of counterexamples to aconjecture by P. Schmid, determine some properties of the possiblecounterexamples of minimum order for each prime, and the isomorphism type ofthe center of the Frattini subgroup for the counterexamples of order 256. Wealso show that nonabelian metacyclic p-groups, nonabelian groups of maximalnilpotency class and 2-groups of coclass two satisfy the conjecture.

On the cohomological triviality of the center of the Frattini subgroup