Let us call the 6 digits D1 to D6.
Given data.
1) D1 + D2 = 7
2) D3 = D1
3) D4 = D2 + 2
4) D4 + D5 = 8
5) D6 = square of D4
6) Sum of D1+D2+D3+D4+D5+D6 = 26
From #5, D6 is a single-digit square number. So it can only be 0,1,4 or 9. Correspondingly, D4 is 0, 1, 2 or 3.
From #3, D4 = D2 + 2. So D4 cannot be 0 or 1. D4 is either 2 or 3
Let's check both the cases
If D4 = 2,
D2 = 0 (from #3) and
D1 = 7 - 0 = 7 (from #1),
D3 = 7 (from #2),
D5 = 6 (from #4) and
D6 = 4 (from #5)
Num is 707264. Sum of digits = 26 which matches.

To confirm, let's try the other possibility also.
If D4 = 3,
D2 = 1(from #3) and
D1 = 7 - 1 = 6 (from #1),
D3 = 6 (from #2),
D5 = 5(from #4) and
D6 = 9 (from #5)
Num is 616359. Sum of digits = 30 which does not match.

Hence the answer is 707264. Solved.

(1) the 6th digit is the square of the 4th:

  x x x 2 x 4  OR  x x x 3 x 9

(2) the 4th digit is the 2nd digit plus 2:
  x 0 x 2 x 4  OR  x 1 x 3 x 9

(3) the sum of the 4th and 5th digits is 8:
  x 0 x 2 6 4  OR  x 1 x 3 5 9

(4) the sum of the 1st and 2nd digits is 7:
  7 0 x 2 6 4  OR  6 1 x 3 5 9

(5) the 3rd digit is equal to the first:
  7 0 7 2 6 4  OR  6 1 6 3 5 9

(6) the sum of the six digits is 26:
  7+0+7+2+6+4 = 26 (OK);  6+1+6+3+5+9 = 30 (not OK)