Artin algebra problem. Taken From Artin algebra  book Page No : $221$ question $2.1$:

Determine the centralizer and order of the  conjugacy class of the matrix 
   $\begin{pmatrix}1 & 0\\0 & 2\end{pmatrix}$ in $GL_2(\mathbb{F}_5)$ 

My attempts : I know  that  centralizer  will be $\begin{pmatrix}a & 0\\0 & a\end{pmatrix},$ where  $a \neq 0$ and $\text{Order}(GL(2,5))=(5^2-1)(5^2-5)=480$
As I'm confused  how can I find the order  of  conjugacy class..
Any hints/solution  will appreciated

I think $$C\Bigg(\begin{pmatrix} 1&0\\0& 2 \end{pmatrix}\Bigg)=\Bigg\{ \begin{pmatrix} a&0\\0& d \end{pmatrix} \in GL(2,\Bbb{F}_5): ad\neq 0 \;\;\text{in} \;\;\Bbb{F}_5\Bigg\}$$
[Check it! If it is wrong, kindly let me know]

Hint for the second one: $$\vert cl(A)\vert=\vert G:C(A)\vert$$ where $cl(A)$ denote the conjugacy class of $A$ and $C(A)$ denote the centralizer of $A$.
