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 p ² and order p ⁿ where p is odd. The number of these is f(n)=p ^{αn 3+o(n 3)} where α=[Formula presented]. For the larger class of all powerful groups of exponent p ² and order p ⁿ, where p is odd, the number is p ^{[Formula presented]n 3+o(n 3)} . Thus here the class of powerfully nilpotent p-groups is large while sparse within the larger class of powerful p-groups.