Classify 2-generated nonabelian p-groups

Classify all 2-generated nonabelian finite p-groups whose subgroup-based generating graph is complete multipartite, thereby completing the structural classification of finite p-groups with complete multipartite subgroup-based generating graphs.

Background

The paper proves that a finite p-group has a complete multipartite subgroup-based generating graph precisely when it is either a direct product of two cyclic p-groups or a 2-generated nonabelian p-group. The first class is explicit, but the second is left in generator-theoretic form rather than being classified up to isomorphism.

A complete classification of the relevant 2-generated nonabelian p-groups would make the characterization more explicit and would complete the group-theoretic description underlying Theorem 3.5. The paper notes that such a classification is technically involved and leaves it for future investigation.

References

Since a complete characterization of $2$-generated non abelian $p$-groups is rather involved, we leave this problem for future investigation.

— Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups  (2610.22268 - Choudhury et al., 15 Oct 2026) in Section 3, subsection “Complete multipartite subgroup based generating graph,” immediately after Proposition 3.10