1,000,000 Cubes

Probability

You have $1,000,000$ cubes of eight different colors, and you arrange them into a $2 \times 2 \times 250,000$ cuboid, so that no two cubes of the same color meet at a side, an edge or a vertex.

Is it possible for any two of the eight corner cubes of the $2 \times 2 \times 250,000$ cuboid to share the same color?