It is the "free" group of nilpotency class at most $2$, exponent dividing $4$, on three generators. (In other words, every group in that class is a quotient of it.)
According to Sergey Malyushitsky (On Sylow $2$-subgroups of Finite Simple Groups of Order up to $2^{10}$, 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 $2$.
