In some kind well-known is that for cycles (and therefore each permutation written as a product of disjoint cycles) the conjugation by another permutation could be easily computed, just replace the symbol in the cycles in concordence with the permutation by which you conjugate.
Now are there similar "shortcut's" for example for other constructions like commutators, powers or generated subgroups? For example for the commutator with $[x,y] = x^{-1}x^y = (y^{-1})^{x} y = (y^x)^{-1}x$ the above shortcut could be applied, but maybe there are other ways too.