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The multiplicity of the irreducible character of $S_n$ indexed by the partition $\lambda$ of $n$ in the action of $S_n$ on itself by conjugation is the coefficient of the Schur function $s_\lambda$ in the Schur function expansion of $1/(1-p_1-p_2-p_3-\cdots)$, where $p_i$ is a power sum symmetric function. (This is an exercise in Chapter 7 of Enumerative Combinatorics, vol. 2.) This makes it very easy to compute the decomposition of the conjugacy action using Stembridge's SF package for Maple. For instance, I computed on my laptop the entire decomposition for $n=20$ in a couple of seconds.