We study the normal Cayley graphs Cay(Sn,C(n,I)) on the symmetric group Sn, where I⊆{2,3,…,n} and C(n,I) is the set of all cycles in Sn with length in I. We prove that the strictly second largest eigenvalue of Cay(Sn,C(n,I)) can only be achieved by at most four irreducible representations of Sn, 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 Cay(Sn,C(n,I)) has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of Sn, and we obtain that Cay(Sn,C(n,I)) does not have the Aldous property whenever n∈I. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of Cay(Sn,C(n,{k})) where 2≤k≤n−2.