NSPACE for checking if two graphs are isomorphic
The algorithm you proposed can be made to work in $O(n \log n)$-space, where $n$ is number of vertices. While permuting the vertices of graph $G_1$, you can also keep the mapping and reverse mapping from and to original vertices. That way you need not rewrite the whole graph. And you can still check if a particular edge in $G_1$ maps to an edge in $G_2$ and vice-versa.

Also, planar graph isomorphism is log-space.