We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida. 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, assuming the tails of the displacement decays at exponential rate; Bérard and Maillard 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.