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I can't sleep, so I spent a stupid amount of time illustrating how a bit mask of 6 bits can address:

1. All 6 faces.
2. All 8 edges.
3. All 8 vertices.

Also, unlike on a NES controller, you can point at two opposing sides (not illustrated).

What are lines that intersect with the opposite faces of a cube at the origin called?

4. All 3 axes.

If what you want to specify is a vertex, not setting the bits for 3 adjacent faces could be used to indicate ambiguity or missing data.