Find $\beta$ as an element of $S_7$ Let $\beta^8 = (1$ $5$ $4$ $3$ $6$), a 5-cycle. Find $\beta$ as an element of $S_7$. 

Any help would be appreciated on this problem. I do not have one similar to this. Thanks!
It is enough to find $\beta$ that is a $5$-cycle.

Then $\beta^{5}=1$ and so $\beta = \beta^{16} = (1 5 4 3 6)^2$.
It is enough to find $\beta$ that is a $5$-cycle.

Then $\beta^{5}=1$ and so $\beta = \beta^{16} = (1 5 4 3 6)^2$.