Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,19,15,18,27,33)(12,16,22,30,21,29,37,40)(13,23,24,34,25,17,26,32)(20,28,36,39,35,31,38,41);(6,7,8,9)(10,12)(11,16)(13,20)(14,22)(15,21)(17,31)(18,29)(19,30)(23,28)(24,36)(25,35)(26,38)(27,37)(32,41)(33,40)(34,39);(10,13,15,25)(11,17,18,23)(12,20,21,35)(14,26,27,24)(16,28,29,31)(19,34,33,32)(22,38,37,36)(30,41,40,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4036);