Calculate the number of different conjugacy classes in $S_6$ and write down a representative permutation for each class. Find an element $g\in S_6$ such that $g(123)(456)g^{-1}=(531)(264)$.
I calculated the number of conjugacy classes in $S_6$, which is 11, and wrote down the permutation for each class, but I dont know how to find the $g$ such that $g(123)(456)g^{-1}=(531)(264)$. So far I tried to do some puzzling to find $g$, but I didn't succeed. I feel like there's a better way to find $g$ than continue to puzzle.
I'd appreciate some tips on how to find $g$!
By luck I found the answer,$g=(25)$, but I'd still like to know the general idea on how to solve this
HINT:
$$g(123)(456)g^{-1} = (g(1),g(2),g(3))(g(4),g(5),g(6))$$