Abstract A finite 𝑝-group 𝐺 is said to be 𝑑-maximal if d ⁢ ( H ) < d ⁢ ( G ) d(H)<d(G) for every subgroup H < G H<G , where d ⁢ ( G ) d(G) denotes the minimal number of generators of 𝐺. A similar definition can be formulated when 𝐺 is acted on by some group 𝐴. In this paper, we extend earlier results of Kahn and Laffey to this more general setting and we answer a question of Berkovich on minimal non-metacyclic 𝑝-groups.

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