2-groups with few conjugacy classes

An old question of Brauer that asks how fast numbers of conjugacy classes g row is investigated by considering the least number c(n) of conjugacy class es in a group of order 2(n). The numbers c(n) are computed for n less than or equal to 14 and a lower bound is given for c(15). It is observed that c( n) grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups G(n). Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computa tions.