We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida in [7]. In this model the size of the population is fixed to a constant N. At each step individuals in the population reproduce independently, making children around their current position. Only the N rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts in [2], assuming the tails of the displacement decays at exponential rate; Bérard and Maillard [3] took interest in the case of heavy tail displacements. We take interest in an intermediate model, considering branching random walks in which the critical spine behaves as an α-stable random walk.