Nilpotent group that is finitely generated as a powered group for a set of primes

Definition
A group $$G$$ is termed a nilpotent group that is finitely generated as a powered group for a set of primes if it satisfies the following equivalent conditions:


 * 1) $$G$$ is a nilpotent group as well as a group that is finitely generated as a powered group for a set of primes. The latter means that there exists a subset $$\pi$$ (possibly empty, possibly finite, possibly infinite, possibly including all primes) such that $$G$$ is a $$\pi$$-powered group and there is a finite subset $$S$$ of $$G$$ such that there is no proper $$\pi$$-powered subgroup of $$G$$ containing $$S$$.
 * 2) $$G$$ is a nilpotent group and there is a set of primes $$\pi$$ such that $$G$$ is a pi-powered group, and further, the abelianization of $$G$$ is finitely generated as a $$\mathbb{Z}[\pi^{-1}]$$-module.