Nullspace Transpose[{{1,1,1,1,1,1,1,1},{4,1,0,-1,-1,1,-1,-1},{5,-1,1,0,0,-1,0,0},{6,0,-2,1,1,0,1,1},{-12,0,0,2,2,0,-2,-2},{-8,2,0,-2,-2,-2,2,2},{-8,-4,0,-2,-2,4,2,2}}]

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