We investigate the properties of clusters of galaxies in the ΛCDM models with a steplike initial power spectrum. We examine the mass function, the peculiar velocities and the power spectrum of clusters in models with different values of the density parameter Ω0, the normalized Hubble constant h and the spectral parameter p, which describes the shape of the initial power spectrum. The results are compared with observations. We also investigate the rms bulk velocity in the models, where the properties of clusters are consistent with the observed data. We find that the power spectrum of clusters is in good agreement with the observed power spectrum of the Abell-ACO clusters, if the spectral parameter p is in the range p = 0.6 − 0.8. The power spectrum and the rms peculiar velocity of clusters are consistent with observations only if Ω0 < 0.4. The Ω0 = 0.3 models are consistent with the observed properties of clusters, if h = 0.50− 0.63. For h = 0.65, we find that Ω0 = 0.20− 0.27.