True or false: there exists a group of order $81$ with trivial center.

Question

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!

Answer 1

notice that $|G:C_G(x_i)|$ is a multiple of $3$ for every non-central $x_i$, so the sum is a multiple of $3$, and hence not $80$.

This can be generalized to finite $p$-groups.