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Exercise (Centralizers Of The Automorphisms Of The Square)
Let $G = D_8$, the automorphisms of the square. Compute the centralizes $C(a)$ for each $a\in G$.
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The centralizers are $$ \begin{align} C(R_0)&=\{R_0,R_{90}, R_{180}, R_{270}, H, V, D, D' \}\\ C(R_{180})&=\{R_0,R_{90}, R_{180}, R_{270}, H, V, D, D' \}\\ C(R_{90})&=\{R_0,R_{90}, R_{180}, R_{270}\}\\ C(R_{270})&=\{R_0,R_{90}, R_{180}, R_{270}\}\\ C(H)&=\{R_0, R_{180}, H, V\}\\ C(V)&=\{R_0, R_{180}, H, V\}\\ C(D)&=\{R_0, R_{180}, D, D'\}\\ C(D-)&=\{R_0, R_{180}, D, D'\}\\. \end{align} $$ Compare this with the exercise about the conjugacy classes of $G$. Do you notice that a large centralizer means a small conjugacy class, and vice versa. The order of the centralizer and the order of the conjugacy class are inversely related. Their product is always the order of the group.