Putative global minima of sodium clusters with up to 380 atoms have been located for two model interatomic potentials in order to identify the structures responsible for the size-dependence of the thermodynamic properties in experiments. Structures based upon the Mackay icosahedra predominate for both potentials, and the magic numbers for the Murrell-Mottram model show excellent agreement with the sizes at which maxima in the latent heat and entropy change at melting have been found in experiment. In particular, the magic numbers at sizes intermediate between the complete Mackay icosahedra are due to unusual twisted icosahedral structures.
