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

Definition
A group $$G$$ is termed a group that is finitely generated as a powered group for a set of primes if 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 $$S$$ is a generating set in the $$\pi$$-powered sense, i.e., there is no proper $$\pi$$-powered subgroup of $$G$$ containing $$S$$.