Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
562 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)(7,11)(9,12)(13,17)(14,18)(16,19)(20,21)(22,23,27,31)(24,28,33,42)(25,35,37,45)(26,38,40,30)(29,39,44,36)(32,46,47,53)(34,50,49,43)(41,51,52,48);(6,7,12,18,10,14,9,11)(8,13,19,21,15,20,16,17)(22,24)(23,28)(25,32)(26,34)(27,33)(29,41)(30,43)(31,42)(35,46)(36,48)(37,47)(38,50)(39,51)(40,49)(44,52)(45,53);(6,8)(7,13)(9,16)(10,15)(11,17)(12,19)(14,20)(18,21)(22,25,27,37)(23,29,31,44)(24,32,33,47)(26,39,40,36)(28,41,42,52)(30,45,38,35)(34,51,49,48)(43,53,50,46);(7,14)(8,15)(11,18)(16,19)(24,33)(28,42)(32,47)(34,49)(41,52)(43,50)(46,53)(48,51)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12560);