How to calculate the number of automorphisms of a given graph? How do determine the number of isomorphisms that a graph has to itself?

For instance, suppose we have the following graph:



How do I determine how many isomorphisms there are from G itself?
Here are all 48 automorphisms generated by Sage:

$$\begin{gathered}
  {\text{() }} \hfill \\
  {\text{(d f)(e g)(h k) }} \hfill \\
  {\text{(d e) }} \hfill \\
  {\text{(a b) }} \hfill \\
  {\text{(b m) }} \hfill \\
  {\text{(f g) }} \hfill \\
  {\text{(a m b) }} \hfill \\
  {\text{(a b)(f g) }} \hfill \\
  {\text{(a b)(d f)(e g)(h k) }} \hfill \\
  {\text{(a b m) }} \hfill \\
  {\text{(b m)(d e) }} \hfill \\
  {\text{(d e)(f g) }} \hfill \\
  {\text{(b m)(d f)(e g)(h k) }} \hfill \\
  {\text{(d g e f)(h k) }} \hfill \\
  {\text{(d f e g)(h k) }} \hfill \\
  {\text{(b m)(f g) }} \hfill \\
  {\text{(a b)(d e) }} \hfill \\
  {\text{(d g)(e f)(h k) }} \hfill \\
  {\text{(a m) }} \hfill \\
  {\text{(a m b)(f g) }} \hfill \\
  {\text{(b m)(d e)(f g) }} \hfill \\
  {\text{(a m b)(d f)(e g)(h k) }} \hfill \\
  {\text{(a b)(d f e g)(h k) }} \hfill \\
  {\text{(a m b)(d e) }} \hfill \\
  {\text{(a b m)(d f)(e g)(h k) }} \hfill \\
  {\text{(b m)(d f e g)(h k) }} \hfill \\
  {\text{(a b m)(d e) }} \hfill \\
  {\text{(a b)(d e)(f g) }} \hfill \\
  {\text{(b m)(d g e f)(h k) }} \hfill \\
  {\text{(a b m)(f g) }} \hfill \\
  {\text{(a b)(d g e f)(h k) }} \hfill \\
  {\text{(a b m)(d g e f)(h k) }} \hfill \\
  {\text{(a b)(d g)(e f)(h k) }} \hfill \\
  {\text{(a m b)(d e)(f g) }} \hfill \\
  {\text{(a m b)(d f e g)(h k) }} \hfill \\
  {\text{(a b m)(d f e g)(h k) }} \hfill \\
  {\text{(a m)(d f)(e g)(h k) }} \hfill \\
  {\text{(a m)(d e) }} \hfill \\
  {\text{(b m)(d g)(e f)(h k) }} \hfill \\
  {\text{(a m b)(d g e f)(h k) }} \hfill \\
  {\text{(a m)(f g) }} \hfill \\
  {\text{(a b m)(d e)(f g) }} \hfill \\
  {\text{(a m b)(d g)(e f)(h k) }} \hfill \\
  {\text{(a m)(d e)(f g) }} \hfill \\
  {\text{(a m)(d f e g)(h k) }} \hfill \\
  {\text{(a b m)(d g)(e f)(h k) }} \hfill \\
  {\text{(a m)(d g e f)(h k) }} \hfill \\
  {\text{(a m)(d g)(e f)(h k)}} \hfill \\ 
\end{gathered} $$
Considering the fact, that isomorphic graph $H$ differs from the an initial graph $G$ by a permutation of columns of adjacency matrix, we can possibly have $n!$ of such permutations, where $n$ is a number of vertices of both graphs.