= Solution
Place the cube's vertices at $(\pm1,\pm1,\pm1)$. Every symmetry fixes its centre, and permutes the three perpendicular pairs of face normals, so it is a <signed permutation matrix>. Conversely every such matrix preserves the cube. The <symmetry group of a cube> acts transitively on its eight vertices: independent coordinate sign changes take $(1,1,1)$ to any vertex.
A symmetry fixing $(1,1,1)$ must permute its three incident edges. This gives an injective map from its <stabilizer subgroup> into $S_3$, since those three edge directions determine the linear transformation. All six permutations are realized by coordinate <permutation matrices>, which fix $(1,1,1)$. Thus the vertex stabilizer has order six. The <orbit-stabilizer theorem> now yields
$$
\boxed{|G|=8\cdot6=48.}
$$
The use of all symmetries includes reflections as well as the twenty-four elements of the <rotational symmetry group of a cube>.
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