I know a few graph invariants and it seemed that these two graphs do not have the same amount of circuit length $3$.
(our definition of a circuit is a closed path, a path being a walk with no repeated edges)
In particular, it seems that $G_1$ has $9$ circuits of length $3$ in the anti-clockwise direction (so there are $18$ circuits of length $3$ in total).
$G_2$ has only $6$ circuits of length $3$ in the anti-clockwise direction so it has $12$ circuits in total of length $3$.
So $G_1$ is not isomorphic to $G_2$, however, my solutions say that they are.