In 1934, Hall introduced regular p-groups. Any regular 2-group is abelian. A regular p-group is power closed, exponent closed, strongly semi-pabelian, and an exact power margin group. The study of these properties and relationships among them constitutes the investigation of the power structure of a p-group. In this paper, we classify 2-generator 2-groups of nilpotency class two according to their power structure.