Give an example or show that no such example exists of the following:
A group of order $81$ with trivial center.
My attempt: Using the class equation, we know that
$|G| = |Z(G)|+\sum_i|G:C_G(x_i)|$.
Since the center is trivial, $|Z(G)| = 1$, so $\sum_i|G:C_G(x_i)| = 80.$ This is really as far as I've been able to go, I'm very stuck here.
Any help would be appreciated!