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Let x denote the rotational symmetries of a cube. The vertex one can be taken to any other vertex by rotation so the orbit of 1 is $$orb_x (1)={1,2,3,4,5,6,7,8}$$ For the stabilisers I have $$stab_x (1)={e,(2,4,5)(3,8,6),(2,5,4)(3,6,8)}$$Where e denotes the identity.

Where do these stabilisers come from?

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    $\begingroup$ It would be useful for you to include a diagram indicating how the vertices of your cube are labeled. $\endgroup$ – Travis Willse May 6 '15 at 18:14
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The fact that the identity is an element of the stabilizer is obvious (since it fixes everything). The other elements of the stabilizer come from rotating the cube along the space diagonal (from vertex $1$ to vertex $7$). There are two such elements because we can rotate in either direction.

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