Small cycles in the star graph

The Star graph is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(1 2),(1 3),\ldots,(1 n)\}$. These graphs are bipartite, they do not contain odd cycles but contain all even cycles with a sole exception $4$-cycles. We characterize all distinct $6$- and $8$-cycles by their canonical forms as products of generating elements. The number of these cycles in the Star graph is also given.