I realized my question here might have been too hard for MSE, so I'm asking it here as well.

The Burnside group $B(d, n)$ is defined as the quotient of the free group on $d$ generators by the normal subgroup generated by all $n$th powers.

**Question.** How do I see that $B(2, 3)$ has $27$ elements and is isomorphic to the group of matrices of the form$$\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$$for $x,y,z\in\mathbb{F}_3$?