Algebraic conjugacy of irreducible characters of a group $GL(2,8)$

The structure of the tables of characters for groups $GL(2,q)$ is known for a long time. However, with setting a specific value for $q$, its finding in explicit form can be very difficult because even calculating numbers, which determine the position of characters in the table, requires considerable effort. It also turns out that specific values of some characters can't be easy for calculating because of nontrivial relations between roots of $1$ of various degrees. In the work a table of the characters of the group $GL(2,8)$, construction of which demonstrated the difficulties above, is presented explicitly. In particular, there are discovered interesting connections between the roots of $1$ degree $21$. Algebraic conjugacy of the characters of the group $GL(2,8)$ is fully defined, which allowed to calculate the rank of the group of central units of the integral group ring of this group.