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We provide a classification of developable cubic hypersurfaces in $\mathbbP^4$. Using the correspondence between forms of degree $3$ on $\mathbb P^4$ andArtinian Gorenstein $\mathbb K$-algebras, given by Macaulay-Matlis duality, wedescribe the locus in ${\rm GOR}(1,5,5,1)$ corresponding to those algebraswhich satisfy the Strong Lefschetz property.

Developable cubics in $\mathbb P^4$ and the Lefschetz locus in ${\rm GOR}(1,5,5,1)$