How many elements are in the conjugacy class of $\tau \in S_9$? Just one simple question:
Let $\tau =(56789)(3456)(234)(12)$.
How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?
First step is to write it in disjunct cyclces I guess. What's next? :)
 A: Looking through my book on group theory (Group Theory by W.R. Scott), I find the following theorems:
Let $n \in \mathbb{N}$, and let $f$ be a function such that $f(i)$ is a nonnegative integer for $1 \le i \le n$.  Let $[f]$ or $[f(1), ..., f(n)]$ denote the set of $x \in \operatorname{Sym}(n)$ such that the cyclic decomposition of x contains $f(i)$ $i$-cycles for $1 \le i \le n$.
Theorem 11.1.1. If $f$ is a function with values nonnegative integers and $\sum_{i=1}^n({f(i) \cdot i)} = n$, then $[f]$ is a conjugate class in $\operatorname{Sym}(n)$.  Conversely, if $x \in \operatorname{Sym}(n)$, then there is a function $f$ such that the conjugacy class of x equals $[f]$.
Theorem 11.1.4. If $[f]$ is a conjugate class in $\operatorname{Sym}(n)$, then $\big| [f] \big| = \frac{n!}{\prod_{i=1}^ni^{f(i)} \cdot f(i)!}$ .

[$Sym(n)$ denotes the group of all permutations of n elements.]
Using these theorems, the answer can easily be found.
A: This is an exercise from BAI (Jacobson) that yields to a combinatorial argument

Let the partition [of the integer $n$] associated with a conjugacy class be $(n_1,n_2,\dots,n_q)$ where $$n_1=\cdots=n_{q_1}>n_{q_1+1}=\cdots=n_{q_1+q_2}>n_{q_1+q_2+1}\cdots$$
Show that the number of elements in this conjugacy class is $$\frac{n!}{\prod q_i!\prod n_j}$$

