In this work the Conformable model, obtained from Fractional Neutron Point Kinetic-model, is extended to consider six groups of delayed neutron precursors. The methodology to determine if the obtained solution is asymptotically stable or unstable of the Conformable model is developed. To show the effect of the functions on the two cases of study (abrupt change of reactivity and reactor start-up), numerical simulations are performed with different anomalous diffusion coefficient values. The results of the Conformable model are equivalent to the results of the classical model. However, the neutron density shows small oscillations around the results of the classical model. The stability analysis is performed through Theorem 1 that establishes the conditions to determine if the solution is asymptotically stable, likewise Theorem 2 is used to determine if the solution is unstable. For both theorems are establish the corresponding inequalities that must be fulfilled.