What rule(s) for a good S-box is this one violating and how does it relate to non-linearity etc.?
I have a software tool for analyzing basic differential and linear properties of s-boxes.
Submitting your s-box to my tool yields the following output:

The most probable differential characteristic(s):
1 -&gt; 215 with probability 10/256 (xor -&gt; xor) (weight 1 -&gt; 6)
Linearity: 18.0 ((166, 166))

Differential probability
So we can see that the most likely difference is that a difference of 1 will output a difference of 215 with probability 10/256. The type of difference used was xor, it is possible to use other functions such as addition to attempt to find other differentials. A probability 10/256 differential is not horrible, but it could be better; For reference, the AES s-box has a max differential probability of 4/256.
Linearity
Linear cryptanalysis is analogous to differential cryptanalysis, but to many it may slightly more complex to reason about. The &quot;linearity&quot; value shows the distance to the nearest linear function. The more closely that the s-box approximates a linear function, the more easily it can be attacked. Again, the value for the s-box you presented is moderately good, but is again not as good as the AES s-box, which has a linearity value of 16.0.
Further Understanding
For a true understanding of the above, I recommend this resource for studying differential and linear cryptanalysis. It is a relatively gentle introduction compared to an actual paper on the subject. I developed my software for analyzing s-boxes while following these tutorials.