Upper bound of group order where $g^3=e \forall g\in G$

Question

If $G$ is a group such that $g^3=e$ for every $g\in G$, what is the upper bound for its order? I am aware of the Heisenberg group, and I cannot find a group with greater order that has this property. So I conjecture that $|G|\leq 27$. As for proving it, I am sure it has something to do with the fact that the order is bounded by $3^3$. Any help is appreciated.

Answer 1

What about the direct product of arbitrarily many copies of $\mathbb{Z}_3$?

Answer 2

If $G$ has exponent $3$ and is generated by $m$ elements, then $|G| \le 3^c$ with $c = m + \binom{m}{2} + \binom{m}{3}$, and this bound is best possible.

According to this site this was proved in 1933 independently by Levi and van der Waerden.