[example: The Conjugacy Classes of $S_3$]
Let $G=S_3$. For every $h \in S_3$,
\begin{align*}
heh^{-1}(x) &= h(e(h^{-1}(x))) \\
&= h(h^{-1}(x)) \\
&= x
\end{align*}
for each $x \in \{1,2,3\}$, so $heh^{-1}=e$ and $\operatorname{Cl}_{S_3}(e)=\{e\}$.
For a transposition $(a\,b)$ and any $h \in S_3$, the conjugate $h(a\,b)h^{-1}$ sends $h(a)$ to $h(b)$, sends $h(b)$ to $h(a)$, and fixes the remaining symbol. Hence
\begin{align*}
h(a\,b)h^{-1}=(h(a)\,h(b)).
\end{align*}
Applying this to $(1\,2)$ gives
\begin{align*}
e(1\,2)e^{-1} &= (1\,2), \\
(1\,3)(1\,2)(1\,3)^{-1} &= ((1\,3)(1)\,(1\,3)(2)) = (3\,2) = (2\,3), \\
(2\,3)(1\,2)(2\,3)^{-1} &= ((2\,3)(1)\,(2\,3)(2)) = (1\,3).
\end{align*}
Thus the conjugacy class of $(1\,2)$ is
\begin{align*}
\{(1\,2),(1\,3),(2\,3)\}.
\end{align*}
For the $3$-cycle $(1\,2\,3)$, the same relabelling rule gives
\begin{align*}
h(1\,2\,3)h^{-1}=(h(1)\,h(2)\,h(3)).
\end{align*}
Taking $h=e$ gives $(1\,2\,3)$, while taking $h=(1\,2)$ gives
\begin{align*}
(1\,2)(1\,2\,3)(1\,2)^{-1}
&= ((1\,2)(1)\,(1\,2)(2)\,(1\,2)(3)) \\
&= (2\,1\,3) \\
&= (1\,3\,2).
\end{align*}
These are the only two $3$-cycles in $S_3$, so their class is
\begin{align*}
\{(1\,2\,3),(1\,3\,2)\}.
\end{align*}
Therefore $S_3$ is partitioned into conjugacy classes of sizes $1$, $3$, and $2$.
[/example]