The center of a group $G$ is
$\{z|z \in G$ and for all $g \in G, gz = zg\}$
Ie...the set of elements that commute with every $g \in G$.
But $gz = zg$ can be written as $g^{-1}zg = z$
So it seems the center of $G$ can also be described as the set of all elements of $G$ that are invariant under conjugation with any element of $G$...is that correct?
If so, it seems that commute and conjugate are the same thing?