The Quillen categories of p-groups and coclass theory

Coclass theory can be used to define infinite families of finite p-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-p cohomology rings. Here we prove that almost all groups in one of these infinite families have equivalent Quillen categories. We also show how the Quillen categories of the groups in an infinite family are connected to the Quillen category of their associated infinite pro-p-group of finite coclass.