Finitely generated and FC implies FZ

Statement
Suppose $$G$$ is a finitely generated group that is a FC-group: every conjugacy class in $$G$$ is finite. Then, $$G$$ is a FZ-group: the center of $$G$$ has finite index in $$G$$.

Proof
The proof idea is that the center is the intersection of centralizers of elements in a generating set. This is an intersection of finitely many subgroups of finite index, hence is finite. Details to be filled in.