Pocket cube!

The eight corners positions: $\displaystyle 8!$

But seven of them can be independently rotated $\displaystyle 3^7$

There is nothing identifying the orientation of the cube in space. This is because all possible positions of the corners are equivalent and no center like $3\times3\times3$ . We can do that by canceling $\times 8$ in $8!$ and $\div3$ in $3^7$

The possible combinations are: $7!\times 3^6= \boxed{3674160}$

(I had to use paper and calculator to solve this)

With eight corner positions, there are a total of 8! possible ways to arrange them, as n number of items can be arranged without repeating itself n! times.

If one of these corners remains in the same place, that leaves seven other corners that can be rotated independently, meaning there are 3^7 or 2187 combinations.

Due to the fixed center (thanks for the hint), there is no initial orientation like a regular Rubik's cube, rotated by any means, resulting in 3 * 8 times less positions .

This reduces down to 7! * 3^6 solutions, or 3674160 ways.

Fun Fact: I could never solve one of these (I'm not the brightest person), so I made one out of 8 6 sided dice and magnets so that you could take apart and solve it if you got frustrated. So worth the time.