I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.

My own calculations are quite slow, even for $n = 8$. I suppose my question is how large could one expect to scale this computation, if one desires an explicit decomposition.

I am aware there are estimates of the parameters known for large $n$ (e.g. see http://www.maths.qmul.ac.uk/~twm/ConjRep.pdf).