A Perazzo algebra is an Artinian Gorenstein algebra whose Macaulay dual generator is a so called Perazzo polynomial. Such algebras never have the strong Lefschetz property. In this talk we focus on Perazzo algebras in at most five variables, in which case the Hilbert function is always unimodal. We will see that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. The possible Jordan types of multiplication by a linear form will also be discussed, in the case of Perazzo algebras with minimal Hilbert function.