2³p阶群的共轭类个数与群的结构
The Number of Conjugate Classes and the Structure of the Group with Order 2³p

在有限群的研究中，群的阶和群的元素的共轭类个数是群的两个非常重要的数量，这两个数量对群的结构性质有很大影响，很多有限群完全可由这两个数量确定.对于阶为2³p（p为奇素数）的有限群，根据分类定理，一共有19类互不同构的2³p阶群，对这19类2³p阶群，利用它们的生成元与生成关系及数论和群论知识确定了它们的共轭类，并由此得出了它们的共轭类的个数，进一步对这19类2³p阶群的共轭类个数进行比较，得到了2³p阶群的共轭类个数的最小值.反过来，根据已得到的2³p阶群的共轭类个数的最小值及群的阶，利用2³p阶群的分类，对19类2³p阶群的共轭类个数进行比较，确定了共轭类个数取最小值的2³p阶群的具体结构.
In the study of finite groups, the order of a group and the number of conjugate classes of the elements of a group are two very important quantities of the group. These two quantities have a great influence on the structure and properties of the group. Many finite groups can be determined completely by these two quantities. For groups of order 2³p (p is an odd prime), the number of conjugate classes of 19 kinds of groups of order 2³p is determined by using their generators and generative relations as well as the knowledge of number theory and group theory. The number of conjugate classes of the 19 kinds of groups of order 2³p is further compared, and the minimum number of conjugate classes of groups of order 2³p is obtained. According to the minimum number of conjugate classes and the order of groups of order 2³p, the number of conjugate classes of 19 groups of order 2³p is compared by using the classification of groups of order 2³p, and the concrete structure of groups of order 2³p with the minimum number of conjugate classes is determined