156726

First I realised that you could only have the same cube on one set of opposing corners. This was because they would hit all the sides and you could put on in the middle (this has no faces visible). I then work systematically to fill in the rest. I worked on the principle that you could have 3, 2 or 1 on show, thus each colour has to add up to 6. I also used 6 different coloured cubes in the middle as my starting point after the first set of diagonal of colours.
I have attached my solution,