[morei]

No, that's exactly the point. Knowing the center of mass and the moment of inertia is sufficient: You don't know need to know the exact shape, or the variations in density inside the object etc etc.

[kergonath]

Yes, the centre of mass and the moment of inertia, i.e., not only the centre of mass. The moment of inertia depends on the shape of the object.

[ndr]

And so does the centre of mass' position...