Is there a site or available program code which can be used to compute the cycle index for various groups, for group $G$ having cycle index $Z(G)$
$$ Z(G)=\frac{1}{|G|}\sum_{g \in G}\prod_{k=1}^{n} a_k^{j_k(g)} $$
i.e. for example the symmetric group has cycle index
$$ Z(S_n)=\sum_{j_1+2j_2+...+nj_n=n} \frac{\prod_{k=1}^{n}a_k^{j_k}}{\prod_{k=1}^{n} k^{j_k} {j_k}!} $$
A lot of websites go straight to Harary's book on Graphical Enumeration, which has a lot of great examples of many groups of which some are tabulated.
In particular it would be interesting to see the result for a cycle index from one of his papers labeled $Z_{n+1}(S_p,1+x)$ which is a cycle index of permutation group of degree $\binom{p}{n+1}$ which is isomorphic to $S_p$, and obtained from $S_p$ by taking sets of $n+1$ objects as the new objects to be permuted, and this was defined to be the counting polynomial for pure $n$-complexes with $p$ points. In his paper he mentioned comparing $Z(S_p)$ to get $Z_2(S_p)$ from certain cycle transformations, but no general outline on how to obtain higher order. If anyone is interested in the specific paper, it is called "The Number of linear, directed, rooted, and connected graphs" where the operation to obtain $Z_2(S_p)$ is described