Eternity II, Part 2

I was thinking about this puzzle the other day ($2 million in prize money is kind of attention getting), and I got to wondering how many different colors/edge patterns you would need in order for all pieces to be different. (The pieces are all square tiles, each edge can be a different color/pattern.) If you have only one color, then there is only one possible pattern: all pieces would be the same. If you used two colors, you could have 16 possible patterns (16 = 2 x 2 x 2 x 2), but rotating pieces will show that some are duplicates of each other. After that it gets complicated, so I wrote a little program to count up the possible patterns and eliminate all the duplicates. The results are in the "picture".

I am thinking there has to be a better way to calculate the number of unique pieces, but I'll be durned if I know what it is. I'll bet Stu knows, or maybe Jack.

Update January 2017 replaced missing image.