729, cube of 9 and square of 27
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For a number to be a perfect square and a perfect cube, it must be a perfect sixth power. Now 36 = 729 is too small and 56 = 15625 is too large, leaving 46 = 4096 as the answer.
It is 64 which is also a square number
If by cube you mean perfect cube (a cube of an integer), then no, and the nearest perfect cube is 81.