Finite p-groups of maximal class with 'large' automorphism groups

The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph g associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph g* of g associated with those p-groups of maximal class whose automorphism group orders are divisible by p - 1. We describe the broad structure of g* by determining its so-called skeleton. We investigate the smallest interesting case p D 7 in more detail using computational tools, and propose an explicit version of Conjecture W for g* for arbitrary p >= 7. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.