A group of order 64 admits a symmetric 2-cocycle with values in C^x that represents a non-trivial cohomology class.
Cohomological and Geometric Approaches to Rationality Problems: New Perspectives, Birkh \"a user Boston, 2010, pp
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Symmetric 2-cocycles with values in $\mathbb{C}^\times$
A group of order 64 admits a symmetric 2-cocycle with values in C^x that represents a non-trivial cohomology class.