Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
494 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)(10,11,13,19,30,34,14,20,16,22,31,35,15,21,28,17)(12,23,24,18,32,36,29,40,27,39,41,38,26,33,25,37);(6,7,8,9)(11,17)(13,28)(14,31)(15,30)(18,33)(19,21)(20,22)(23,37)(24,25)(26,32)(29,41)(34,35)(36,38)(39,40);(10,12)(11,18)(13,29)(14,24)(15,32)(16,27)(17,33)(19,39)(20,36)(21,40)(22,38)(23,35)(25,31)(26,30)(28,41)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6010);