Let $g$ be a generator of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ then the ordered list $$(g^1, g^2, ..., g^{p-1})$$ is a permutation of ordered list $$(1,2,...,p-1)$$ From playing around with small primes, the permutation seems like a random permutation. Is there a rigorous explanation for this pseudo-randomness or a way to measure how random the permutation is? If so, do the results hold for finite cyclic groups in general?
Example: $g=251\in (\mathbb{Z}/257\mathbb{Z})^{\times}$
(251,36,41,11,191,139,194,121,45,244,78,46,238,114,87,249,48,226,186,169,14,173,247,60,154,104,147,146,152,116,75,64,130,248,54,190,145,158,80,34,53,196,109,117,69,100,171,2,245,72,82,22,125,21,131,242,90,231,156,92,219,228,174,241,96,195,115,81,28,89,237,120,51,208,37,35,47,232,150,128,3,239,108,123,33,59,160,68,106,135,218,234,138,200,85,4,233,144,164,44,250,42,5,227,180,205,55,184,181,199,91,225,192,133,230,162,56,178,217,240,102,159,74,70,94,207,43,256,6,221,216,246,66,118,63,136,212,13,179,211,19,143,170,8,209,31,71,88,243,84,10,197,103,153,110,111,105,141,182,193,127,9,203,67,112,99,177,223,204,61,148,140,188,157,86,255,12,185,175,235,132,236,126,15,167,26,101,165,38,29,83,16,161,62,142,176,229,168,20,137,206,49,220,222,210,25,107,129,254,18,149,134,224,198,97,189,151,122,39,23,119,57,172,253,24,113,93,213,7,215,252,30,77,52,202,73,76,58,166,32,65,124,27,95,201,79,40,17,155,98,183,187,163,50,214,1)