Log cycles in $GF(2^8)$

Question

I'm tabulating logs in $GF(2^8)$.

I noticed

1. there's one fixed-point: $\log(235) = 235$
2. there's one 3-cycle: [86, 219, 177]
3. there's one 9-cycle: [15, 75, 179, 171, 178, 211, 82, 148, 38]
4. the rest is one "flight" of length 242 ending in [...,4,2,1,0]

I wonder if

1. it's obvious
2. it's useful
3. it can be found in other structures