  We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian
Gorenstein algebras associated to Perazzo hypersurfaces in projective space.
The main focus lies on Perazzo threefolds, for which we prove that the Hilbert
functions are always unimodal. Further we prove that the Hilbert function
determines whether the algebra is weak Lefschetz, and we characterize those
Hilbert functions for which the weak Lefschetz property holds. By example, we
verify that the Hilbert functions of Perazzo fourfolds are not always unimodal.
In the particular case of Perazzo threefolds with the smallest possible Hilbert
function, we give a description of the possible Jordan types for multiplication
by any linear form.
