On the 2-Generator p-Groups with Non-cyclic Commutator Subgroup

A complete classification of $p$-groups of every nilpotency class is given by R.J. Miech in 1975 where the commutator subgroup is cyclic. The attentions of M.R. Bacon in 1993 and since then L.-C. Kappe in 1999 on the study and the classification of 2-generated $p$-groups based on the nilpotency 2 groups which have the cyclic commutator subgroups. In this paper, we attempt to study the finite 2-generator $p$-groups of nilpotency class 3, where the commutator subgroup is non-cyclic, and identify the structure of one class of such $p$-groups for every prime $p\neq 2,3$.