Bounds in groups with trivial Frattini subgroup

Let G be an arbitrary group with finite commutator subgroup G′. We prove that if the Frattini subgroup Φ(G) is trivial then |G:Z(G)|⩽2|G′|, and this estimate is sharp. A special case of this result was conjectured by M. Herzog, G. Kaplan and A. Lev.