It is the "free" group of nilpotency class at most 2, exponent at most 4, on three generators. (In other words, every group in that class is a quotient of it.)
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.
In general, the "free" group of nilpotency class at most 2 and exponent at most 4 on d generators has order $4^d\cdot 2^{\binom{d}{2}}$. So, for example, you should see a similar phenomena at order 32, where there is a unique group of rank 2 and $p$-class $2$.