Metabelian p-groups and coclass theory

We show that the coclass tree associated with the metabelian p-groups of a fixed coclass is virtually periodic. The proof of this result is obtained by showing that the metabelian p-groups of a fixed coclass consist of finitely many coclass families and finitely many other groups. This also implies that the metabelian p-groups of a fixed coclass can be classified in the sense that they can be described by finitely many parametrised presentations.