Show that $\operatorname{Gal}((x^{8}-1)/(x-1))$ is abelian.

Question

Here ( Prove that the Galois group of $x^n-1$ is abelian over the rationals ) there is a proof that $\operatorname{Gal}(x^{n}-1)$ is abelian, but I'm having trouble connecting it to my situation. Any suggestions are appreciated!