Reduced fusion systems over 2-groups of small order

Kasper K. S. Andersen, Bob Oliver, and Joana Ventura

We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced 
fusion system over $S$ is the fusion system of a finite simple group and is 
tame. It then follows that each saturated fusion system over a $2$-group of 
order at most $2^9$ is realizable. What is most interesting about this result is 
the method of proof: we show that among $2$-groups with order in this 
range, the ones which can be Sylow $2$-subgroups of finite simple groups 
are almost completely determined by criteria based on Bender's 
classification of groups with strongly $2$-embedded subgroups.

Status: submitted