Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#18032 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C23×C5 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,32)(25,29)(26,38)(27,30)(28,34)(31,40)(33,46)(35,45)(36,44)(37,50)(39,42)(41,48)(43,49)(47,53)(51,52);(22,24,27,35)(23,28,30,42)(25,33,36,47)(26,39,40,34)(29,41,44,52)(31,45,38,32)(37,51,49,48)(43,53,50,46);(6,8,9,15,10,14,16,20)(7,11,12,18,13,17,19,21)(22,25,27,36)(23,29,30,44)(24,33,35,47)(26,37,40,49)(28,41,42,52)(31,43,38,50)(32,46,45,53)(34,48,39,51);(8,14)(11,17)(15,20)(18,21)(23,30)(25,36)(28,42)(31,38)(32,45)(33,47)(37,49)(48,51)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11020);