TITLE: How do you calculate the number of elements conjugate to a particular permutation?
QUESTION [8 upvotes]: Working in $S_n$, and given a particular permutation, how would you go about calculating the number of conjugate elements to the permutation?
I guess since two elements are conjugate iff they have the same cycle type this gives some restriction on the conjugate elements. I know two elements, say $x$ and $y$, are conjugate if $g^{-1}\cdot x\cdot g=y$ for some $g$ in the group but I'm not sure how to work out the number of conjugate elements. 
As an example, the permutation is $x = (12)(34)(56789)$, working in $S_{12}$. So $x$ has cycle type $[2,2,5,1,1,1]$ so clearly any conjugate elements must have cycle type $[2,2,5,1,1,1]$.
Any help would be greatly appreciated. Thanks.

REPLY [5 votes]: For the given example, you just have to count the possible $[2,2,5]$ groupings of $12$ elements. Note that if we exchange the two $2$-cycles, we get the same permutation.
Now it can go like: first choose $2$ elements out of $12$, then $10$ elements are left, choose $2$ again, and then choose $5$ elements out of the rest ($8$), finally divide it by $2!$ because the order of the first two chosen group doesn't matter, that is:
$$\frac1{2!}\cdot\binom{12}2\cdot \binom{10}2\cdot\binom85$$