Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19060 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
822 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)(8,14)(9,16)(11,19)(12,17)(15,18)(20,21)(22,23,28,33)(24,34,37,46)(25,38,39,47)(26,32,27,31)(29,41,44,36)(30,42,45,40)(35,43,49,53)(48,50,52,51);(6,7,10,13)(8,11,15,20)(9,17,16,12)(14,21,18,19)(22,24)(23,29)(25,35)(26,41)(27,36)(28,37)(30,43)(31,46)(32,34)(33,44)(38,52)(39,49)(40,51)(42,50)(45,53)(47,48);(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,16)(17,19)(22,25)(23,30)(24,35)(26,40)(27,42)(28,39)(29,43)(31,38)(32,47)(33,45)(34,48)(36,50)(37,49)(41,51)(44,53)(46,52);(25,39)(30,45)(35,49)(38,47)(40,42)(43,53)(48,52)(50,51)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12048);