On one representation of elements of finite $2$-groups in the form of Boolean vectors

In this paper, we propose a way to represent elements of finite $2$-groups as Boolean vectors. Let $G$ be some finite (Burnside) 2-group and its order is $2^k$. In this case, each element of the group will be represented by a unique Boolean (bit) vector of dimension $k$. To calculate the product of two elements, we use analogues of Hall polynomials but now instead of multiplication and addition over the field $\mathbb{Z}_2$ we use the equivalent Boolean (bitwise) operations “and”, as well as “exclusive or”. Note that operations on bits are much faster on a computer than on integer or string data types. For problems requiring the calculation of a large number of products of group elements the method will dramatically reduce the running time of computer programs.