As can be seen from the figure, because it is a cube, the area increased by each cut is equal whether it is horizontal or vertical. Therefore, the sum of the surface areas of 24 cuboids = original area+increased area, and the increased area per cut =6X6X2.

So: S = 6x6x6+6x2x6 = 6x6x3x3 = 648 (square centimeter).

Note: If the cutting method according to the diagram is correct, then the last 24 basic bodies can't be cubes, only cuboids.