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×#18744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
852 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)(8,14)(9,11)(10,18)(12,16)(13,20)(15,19)(17,21)(23,28)(24,31)(25,26)(30,34)(32,33)(35,36);(6,8,11,17)(7,12,9,15)(10,19,20,16)(13,21,18,14)(22,23,27,30)(24,29,33,37)(25,28,26,34)(31,35,32,36);(6,9,11,7)(8,15,17,12)(10,18,20,13)(14,16,21,19)(22,24,27,33)(23,29,30,37)(25,31,26,32)(28,35,34,36);(7,9)(12,15)(13,18)(14,21)(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, 11732);