On conjugacy classes of $S_n$ containing all irreducibles

It is shown that for the conjugation action of the symmetric group $S_n,$ when $n=6$ or $n\geq 8,$ all $S_n$-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition $\lambda$ of $n$ with at least two parts, whose parts are all distinct and taken from the set of odd primes and 1. The following simple characterisation of conjugacy classes containing all irreducibles is proved: If $n\neq 4,8,$ the partition $\lambda$ of $n$ indexes a global conjugacy class for $S_n$ if and only if it has at least two parts, and all its parts are odd and distinct.