Finite simple groups with Sylow 2-subgroups of order $2^7$

The following theorem is proved in the paper. If a Sylow 2-subgroup $T$ of a finite simple group is of order $2^7$ , then either the nilpotency class of $T$ is not greater than 2 or the sectional 2-rank of $T$ does not exceed 4. This theorem and known classification results lead to a list of all finite simple groups with Sylow 2-subgroups of order $\leqslant2^7$.
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