Monte Carlo simulations of a two-dimensional hard homonuclear dumbbell-shaped-molecules system with a high anisotropy parameter ${\mathit{d}}^{\mathrm{*}}$=0.924 were performed for three different thermodynamic states: (i) fluid; (ii) aperiodic molecular solid, whose molecular arrangements can be represented by certain decorations of the kagome$iaa— lattice; and (iii) high-density solid, whose molecular structure corresponds to dense packing of the dumbbell-shaped molecules. The latter structure was exemplified in this study by a herring-bone-type crystal. Explicit calculations of the free energies of the different states, together with the analysis of the melting region, show that the aperiodic solid structures form a thermodynamically stable phase between the fluid and the dense solid. The equation of state, structural properties, and the locations of the phase transitions are reported. The role of the degeneracy entropy in the aperiodic phase is discussed.

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