We study the complexity of multiplication in noncommutative group algebras
which is closely related to the complexity of matrix multiplication. We
characterize such semisimple group algebras of the minimal bilinear complexity
and show nontrivial lower bounds for the rest of the group algebras. These
lower bounds are built on the top of Bl\"aser's results for semisimple algebras
and algebras with large radical and the lower bound for arbitrary associative
algebras due to Alder and Strassen. We also show subquadratic upper bounds for
all group algebras turning into "almost linear" provided the exponent of matrix
multiplication equals 2.