  We study the normal Cayley graphs $\mathrm{Cay}(S_n, C(n,I))$ on the
symmetric group $S_n$, where $I\subseteq \{2,3,\ldots,n\}$ and $C(n,I)$ is the
set of all cycles in $S_n$ with length in $I$. We prove that the strictly
second largest eigenvalue of $\mathrm{Cay}(S_n,C(n,I))$ can only be achieved by
at most four irreducible representations of $S_n$, and we determine further the
multiplicity of this eigenvalue in several special cases. As a corollary, in
the case when $I$ contains neither $n-1$ nor $n$ we know exactly when
$\mathrm{Cay}(S_n, C(n,I))$ has the Aldous property, namely the strictly second
largest eigenvalue is attained by the standard representation of $S_n$, and we
obtain that $\mathrm{Cay}(S_n, C(n,I))$ does not have the Aldous property
whenever $n \in I$. As another corollary of our main results, we prove a recent
conjecture on the second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,\{k\}))$
where $2 \le k \le n-2$.
