On imaginary abelian number fields of type (2, 2,..., 2) with one class in each genus

We show that there exist only finitely many imaginary abelian number fields of type (2,2,...,2) with one class in each genus. Moreover, if the Generalized Riemann Hypothesis is true, we have exactly 301 such fields, whose degrees are less than or equal to 2³. Finally we give the table of those 301 fields.