Simple group problem. I am reading a book of algebra and I have a difficulty in understanding one thing.
In the book it is written there is no simple group of order 528 and the explanation is as follows.
Let $H$ be a Sylow 11-subgroup. Then $n_{11}=12$, which I understood why, and $|N(H)|=44$ and $G$ is isomorphic to a subgroup of $A_{12}$.
Could you please explain?

There is a group of order 528 - in fact, there are 170 of them up to isomorphism. 
See here:
https://groupprops.subwiki.org/wiki/Groups_of_order_528
