Epicentral series

Definition
The epicentral series (sometimes also called the upper epicentral series) of a group is an ascending series defined as follows. $$Z_0^*(G)$$ is the trivial subgroup, and for a positive integer $$i$$, $$Z_i^*(G)$$ is

In particular, $$Z_1^*(G) = Z^*(G)$$ is the epicenter of $$G$$. Note, however, that the epicentral series does not satisfy $$Z^*(G/Z_{i-1}^*(G)) = Z_i^*(G)/Z_{i-1}^*(G)$$.