Group of prime-sixth or higher order contains abelian normal subgroup of prime-fourth order for prime equal to two

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Any group of order 2^n, n \ge 6contains an Abelian normal subgroup (?) (hence, Abelian normal subgroup of group of prime power order (?)) of order 16 = 2^4. In other words, for the prime p = 2, any group of order p^n, n \ge 6 contains an abelian normal subgroup of order p^4.

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