Fixed-class capable group

Definition
Suppose $$c$$ is a positive number. A group $$G$$ is a class $$c$$-capable group if there exists a group $$K$$ such that $$G \cong K/Z_c(K)$$ where $$Z_c(K)$$ denotes the $$c^{th}$$ member of the upper central series of $$K$$.

Note that $$Z_1(K) = K$$ is the center and class 1-capable group is the same notion as capable group.