It is the free group of nilpotency class at most 2, exponent at most 4, on three generators.

According to 

https://etd.ohiolink.edu/rws_etd/document/get/osu1086112148/inline

(ON SYLOW 2-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO 2^10, Sergey Malyushitsky, M.S. Thesis, The Ohio State University)

it is not the Sylow 2-subgroup of a finite simple group.