Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
808 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)(24,34)(25,38)(26,33)(27,32)(28,31)(29,36)(30,39)(35,43)(37,46)(40,47)(41,44)(42,45)(48,50)(49,52)(51,53);(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)(31,44)(32,46)(33,34)(38,52)(39,50)(40,51)(42,49)(45,53)(47,48);(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,16)(17,19)(22,25,28,40)(23,30,31,45)(24,35,37,51)(26,39,27,42)(29,43,44,53)(32,38,33,47)(34,48,46,52)(36,49,41,50);(23,31)(29,44)(30,45)(32,33)(34,46)(38,47)(43,53)(48,52)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11991);