This variation on the Rubik's cube is called the Magic Octagonal Prism, or just The Barrel or The Octagon. In Dutch it is called 'Peperbus', which means pepperpot. It is exactly the same as the normal cube except that the four vertical edges of the cube are truncated. The corner pieces of original cube now have only two colours and the four edge pieces in the middle layer have only one colour.
The differences make this puzzle a little confusing. It is possible to get to a position where it seems that two pieces need to be swapped or where one edge piece needs to be flipped, both of which are impossible on the original cube.
This leaves 8!·12!·37·28/4! = 450,541,700,775,936,000 or 4.5·1017 positions.
Use any solution method for the Rubik's Cube to solve it as far as possible. As mentioned above, there are two situations that may occur that cannot happen on a normal cube.
Some people prefer to hold the puzzle so that the last layer they solve is one with two shaved edges (e.g. the octagonal sides are at L and R). This way any final flips/swaps only involve pieces of the final layer.