From point 2, its clear that first digit is either 1 or 8.. (cube of 1 or 2) (else the cube of 3 or more will not be a single digit). But 5th point says none of the digits repeat, so first digit is 8 and second digit is 2.
So, as per point 1, third digit is 7. (as second digit is 2).
As per point 4, second-to-last digit is none other that 4th digit itself for a 5 digit number. So its equal to 6 times of itself. Means, its 0.
the remaining 5th digit is obtained from point 3. the smallest number with exact five divisors is 12 (2,3,4,6,12). So the fifth digit is (12-3rd digit) = 12-7 = 5.
So the number is 82705.