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 = 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.