On an embedding of certainp-groups

It is shown that ap-group with cyclic centre can be embedded in a finite group as a normal subgroup contained in its Frattini subgroup if and only if it is either an extraspecial 2-group of order at least 2⁷ or the central product of a cyclic groupQ of order ≧4 and an extraspecial groupE of order ≧2⁵, amalgamating Ω₁ (Q) and the centre ofE.