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We studied the formation of cluster crystals in one-dimensional and two-dimensional systems, with Brownian particles interacting via soft-core repulsive potential, of stretched exponential type with exponent α. This model is called the generalized exponential model (GEM-α). Such model has very interesting features, because although it has an exclusively repulsive behavior for any distance, for low temperatures the model forms spatial patterns due to the aggregation of particles, forming periodic crystals of particle agglomerates. We analyze the behavior of the system through numerical simulations of discrete particles using molecular dynamics techniques. We measure dynamic and static properties, always trying to establish a parallel between the physical observable and the quantities needed to understand the system. In one dimension, we found evidence of a thermodynamic phase transition for certain values of α. In the two-dimensional system, we have studied the different cluster-crystalline ground states as a function of model parameters and density. We have also measured different spatial correlation functions in order to characterize the nature of phases, either crystalline, hexatic or liquid. The nature and the number of transitions between cluster-crystal and liquid phases are also investigated, showing different possible sequence of transitions that may explain how meltdown occurs in this two-dimensional system.