Is the symmetric group over 8 items, $S_8$, presented by $\langle x, y \mid x^2 = e, y^8 = e, (xy)^7 = e \rangle$?
Let's consider $S_8$ and let $x = (12)$ and $y = (12345678)$. Then it's self-evident that $x^2 = e$ and $y^8 = e$. Furthermore, $xy = (2345678)$, and so $(xy)^7 = e$. Therefore, $S_8$ is generated (non-freely) by $x$ and $y$, and satisfies the three equations in the presentation.
However, I don't know how to show that $S_8$ doesn't satisfy any equations that can't be proved from the above. In other words, I don't know how to show that if $E$ is an expression written using $x$ and $y$, and the equation $E = e$ is true in $S_8$ using the definitions above, then the equation $E = e$ can also be proved from the three equations in the presentation. On the face of it, that seems like quite a complicated task, and I don't have any ideas for how to make it simpler.
How can this be shown (or is it even true at all)?
(Is this easily extended to show that for all positive integers $n$ (including $n = 1$), $S_n$ is presented by $\langle x, y \mid x^2 = e, y^n = e, (xy)^{(n-1)} = e \rangle$?)