[GAP Forum] Question about symmetric group of nine letters

[Hamid Shahverdi]

Dears,
How can one try by gap to show that

If a centerless group G of order 9! has the following properties, then it
is isomorphic to S_9
1) size of all conjugacy classes are equal to the one in S_9.
2) the order of all maximal abelian subgroups are equal to the one in S_9.


Thanks in advance for any help.
Best Regards
Hamid Shahverdi