Conclaves of planes in PG (4,2) and certain planes externalto the Grassmannian ${cal G}_{1,4,2} subset$ PG(9,2)

We show that in $operatorname{PG}(4,2)$ there exist octets $mathcal{P}_{8}={pi_{1},,ldots,,pi_{8}}$ of planes such that the 28intersections $pi_{i}cappi_{j}$ are distinct points. Suchconclaves (see [6]) $mathcal{P}_{8}$ of planesin $operatorname{PG}(4,2)$ are shown to be in bijective correspondencewith those planes $P$ in $operatorname{PG}(9,2)$ which are external tothe Grassmannian $mathcal{G}_{1,4,2}$ and which belong to the orbit$operatorname{orb}(2gamma)$ (see [4]). The factthat, under the action of $operatorname{GL}(5,2),$ the stabilizergroups $mathcal{G}_{mathcal{P}_{8}}$ and $mathcal{G}_{P}$ both havethe structure $2^{3}:(7:3)$ is thus illuminated. Starting out from aregulus-free partial spread $mathcal{S}_{8}$ in$operatorname{PG}(4,2)$ we also give a construction of a conclave ofplanes $Pinoperatorname{orb}(2gamma)subsetoperatorname{PG}(9,2).$