Context. Radiative transfer plays a major role in the process of star formation, the details of which are still not entirely understood. Aims: Many previous simulations of gravitational collapse of a cold gas cloud followed by the formation of a protostellar core have used a grey treatment of radiative transfer coupled to the hydrodynamics. However, the dust opacities that dominate circumstellar extinction show strong variations as a function of frequency. In this paper, we used a frequency-dependent formalism for the radiative transfer to investigate the influence of the opacity variations on the properties of Larson's first core. Methods: We used a multigroup M1 moment model for the radiative transfer in a 1D Lagrangean Godunov radiation hydrodynamics code to simulate the spherically symmetric collapse of a 1 M&sun; cold cloud core. Monochromatic dust opacities for five different temperature ranges were used to compute Planck and Rosseland means inside each frequency group. Results: The results are very consistent with previous studies and only small differences were observed between the grey and multigroup simulations. For a same central density, the multigroup simulations tend to produce first cores with a slightly higher radius and central temperature. We also performed simulations of the collapse of a 10 and 0.1 M&sun; cloud, which showed that the properties of the first core (size, mass, entropy, etc.) are independent of the initial cloud mass, with again no major differences between grey and multigroup models. Conclusions: For Larson's first collapse, where temperatures remain below 2000 K, the vast majority of the radiation energy lies in the infrared regime and the system is optically thick. In this regime, the grey approximation satisfactorily reproduces the correct opacities, as long as there are no strong opacity variations on scales much smaller than the width of the Planck function. However, the multigroup method is expected to yield additional more important differences in the later stages of the collapse when high-energy (UV and X-ray) radiation is present and matter and radiation are strongly decoupled.