Determine the number of symmetries in the following graph:

![enter image description here](https://i.stack.imgur.com/B2DVG.jpg)

What are the general things you should do when finding such symmetries? Usually I would label all of them from $1...n$ and note their valency mentally

I am still not clear on how you would progress though. I thought that I could permute the point 1 to any of the other points with valancy three, but I have been told you can't, although I am still not sure why. 

Could someone explain this?

So taking the fact that the point 1 has to be fixed, it was then said there were $3!$ permutations of the outer points (the points branching from point 1 (2,5,8)) and for each of those outer points, $2!$ permutations of the points attached to them.(i.e for point 2, there would be $2!$ permutations of 3 and 4)

Why is it that, for example, points 3 and 4 can only be rearranged and that 3 could not be permuted to point 10 for example. They both have the same valency, I am just a bit unclear on why you can't.

Many thanks.