On the classification of prime-power groups by coclass

By giving a constructive proof of Conjecture P of Newman and O'Brien, we reduce the classification of certain p-groups of coclass r to a finite calculation. For the case p = 2 we show that all 2-groups of coclass r can be classified by finitely many paxametrised presentations. A non-constructive proof of Conjecture P was given by du Sautoy, using the theory of zeta functions. Our constructive proof uses homological algebra. It yields more precise results and detailed structure theorems for the p-groups under consideration.