## automorphism group of the Petersen graph

My question is how do you show the automorphism of the Petersen graph is $S_5$ ?

Can you actually say that by drawing it, you can see that when you contract the 'border' vertices to the 'inner' vertices of the Petersen graph, you get $K_5$ which has $S_5$ as its automorphism group. Or is there another way of explaining this?