Let $\sigma=(123)(58)$ in the symmetric group $S_{8}$ of degree 8.
find the order of sigma
I use sum of least squares = $3*2$ =6
So the order is 6
second part is find a $\tau \in S_{8}$ such that $\tau\sigma\tau^{-1}=\sigma^{5}$
I know the order is 6 so $\sigma^{5}=\sigma^{-1} =(132)(58)$.
I can't really figure out how to find $\tau$ with cycle decomposition.
I know $\tau\sigma\tau^{-1}=(321)$ by sending
$ \tau(1)=3, \tau(2)=2,\tau(3)=1$ But why I thought that I should send it to (132) as $\sigma^{5}=\sigma^{-1} =(132)(58)$.
Please give me help to simplify
Conjugation in a symmetric group amounts to relabling of the elements the group acts on. You have $\sigma=(123)(58)$, and you want, through conjugation, to make it into $\sigma^5=(132)(58)$. Easy: what you want is to swap the labels of $2$ and $3$, and not really touch any other labels. The simplest permutation that does this is $(23)$ so we set that to be our $\tau$.
You can check yourself that $$(23)(123)(58)(23)^{-1}=(132)(58)$$.
There are many other relablings that would work, of course. Basically any order-2 permutation on $\{1,2,3\}$ will do the trick, as $(132)=(213)=(321)$. Adding in the relabling $5\leftrightarrow 8$ also wouldn't change anything. And neither would any relabling of the three elements that $\sigma$ doesn't touch, i.e. $\{4,6,7\}$.
Fair warning: I can never remember if you want your relabling permutation to be $\tau$ or $\tau^{-1}$. Fortunately, in this case it doesn't matter. If you encounter an example where the relabling you want is not a transposition, then you will have to try both ways.