There exists a 2-group with a maximal elementary abelian subgroup of order four and rank four

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Statement

There exists a finite 2-group P having a maximal elementary abelian subgroup A of order four (i.e., a Klein four-group that is not contained in any bigger elementary abelian subgroup), and such that P has an elementary abelian subgroup B of rank 4 (i.e., of order 16). In other words, for the prime two, there can be a Group of prime power order having a maximal elementary abelian subgroup of prime-square order (?) that has rank four.

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