Powerfully nilpotent groups

We introduce a special class of powerful p-groups that we call powerfully nilpotent groups that are finite p-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an ‘ancestry tree’ and powerful coclass. We show that there are finitely many powerfully nilpotent p-groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent p² and order pⁿ where p is odd. The number of these is f(n)=p^αn3+o(n3) where α=[Formula presented]. For the larger class of all powerful groups of exponent p² and order pⁿ, where p is odd, the number is p^{[Formula presented]n3+o(n3)}. Thus here the class of powerfully nilpotent p-groups is large while sparse within the larger class of powerful p-groups.