On prime-power groups with two generators

Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of G will be written For any subgroup H of G we denote by P ( H ) the subgroup of H generated by all elements x p as x runs through H , and by Φ( H ) the Frattini subgroup of H . We write ( H :Φ( H )) = p d(H) ; thus d ( H ) is the minimal number of generators of H .