Epinilpotent group

Definition
A group $$G$$ is termed epinilpotent if there is a nonnegative integer $$c$$ such that it satisfies the following equivalent conditions:


 * 1) The class c-epicenter of $$G$$ equals the whole group $$G$$.
 * 2) The only quotient group of $$G$$ that is a class c-capable group is the trivial group.

The smallest $$c$$ satisfying any of these equivalent conditions is termed the epinilpotency class.

Note that finitely generated and epinilpotent implies cyclic, so among finitely generated groups, the only epinilpotent group are cyclic, hence epabelian, so they have epinilpotency class one.