Cascading　failures　on　interdependent　networks　has　attracted　much　attention　in　the　field　of　network　science.　However,　the　effects　of　the　distribution　of　group　size　on　cascading　failures　still　puzzled　us.　In　this　paper,　we　propose　a　cascading　failure　model　on　two-layered　partially　interdependent　networks　with　groups,　in　which　all　nodes　in　the　same　group　are　functional　dependent　on　each　other.　A　percolation　theory　is　developed　to　investigate　the　final　giant　component,　and　the　theory　well　agrees　with　numerical　simulations.　Through　extensive　numerical　simulations　and　theoretical　analyses,　we　find　a　discontinuous　phase　transition　in　the　system.　In　addition,　the　final　giant　component　increases　with　the　heterogeneity　of　group　size　distribution,　and　the　percolation　threshold　decreases　with　the　heterogeneity　of　group　size　distribution.　Furthermore,　the　phase　transition　is　discontinuous　for　relatively　strong　dependency,　while　the　phase　transition　is　continuous　for　relatively　weak　dependency.　The　degree　distribution　of　the　interdependent　networks　does　not　qualitatively　alter　the　above　presented　phenomena.　(C)　2019　Elsevier　B.V.　All　rights　reserved.