The democratic mapping is used for the calculation of low-lying states of nuclei in the sd and fp shells. In addition to demonstrating the applicability of the method in realistic cases where many non-degenerate levels are present, the method allows for the ranking of the various bosons according to their importance as building blocks of low-lying states. It is proven that the s and d bosons are the most important building blocks, followed by the d' and g bosons. Thus one of the basic assumptions of the interacting boson model (IBM) is proven to be correct. Very good agreement between the boson calculation and the shell-model results is obtained for A=20 nuclei when 12 bosons are taken into account, while an even larger number of bosons is required to reproduce the low-lying states of the A=44 nuclei. In order to obtain equally good results with a smaller number of bosons one needs to introduce effective boson Hamiltonians which correspond to truncated fermion spaces.