What kind of fraction will have this property? List out i/19: 

They form a square, and the sum of each column is the same, 81.
i/7 also have such property:
142857
285714
428571
571428
714285
857142

Any other fractions have such property?
What kind of fractions will have such property?

You are after numbers that are the decimal period of so-called cyclic numbers.
$7$ and $19$ in this context are called full reptend primes (OEIS A001913): 

Primes p such that the decimal expansion of 1/p has period p-1, which
  is the greatest period possible for any integer.

The first few are $7, 17, 19, 23, 29, 47, 59, 61, 97.$
