Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,17)(15,20)(16,18)(19,21)(23,28)(24,31)(25,26)(30,34)(32,33)(35,36);(7,11)(9,17)(12,18)(13,14)(16,21)(19,20)(22,23)(24,29)(25,28)(26,34)(27,30)(31,35)(32,36)(33,37);(6,8,10,15)(7,12,13,20)(9,16,17,21)(11,18,14,19)(22,24,27,33)(23,29,30,37)(25,31,26,32)(28,35,34,36);(7,13)(8,15)(11,14)(16,21)(22,25,27,26)(23,28,30,34)(24,32,33,31)(29,36,37,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11730);