Abstarct: Let be a finite -group. The number of non-central conjugacy classes of a given size in is divisible by . Naturally it is better to investigate the groups in which this number is exactly . Macdonald has shown that if is a -group of order and has classes of size , then , , and . The main point of this thesis is baxsed on the non-central classes of minimal breadth. We investigate groups in which the number of classes of minimal size is exactly . In chapter 1, we present basic and preliminary contexts. In chapter 2, we study -groups of maximal class. In chapter 3, we investigate mextabelian groups and groups of maximal class with recent property and at the end we classify of CF mextabelian groups and mextabelian groups of maximal class.