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×#19063 |
| 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 | |
806 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)(25,39)(26,31)(27,32)(28,33)(29,37)(30,41)(35,43)(36,44)(38,47)(40,45)(42,46)(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)(26,36)(27,37)(28,38)(30,43)(31,34)(32,47)(33,44)(39,52)(40,49)(41,50)(42,51)(45,53)(46,48);(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,16)(17,19)(22,25,28,42)(23,30,33,45)(24,35,38,51)(26,41,27,40)(29,43,44,53)(31,46,32,39)(34,48,47,52)(36,50,37,49);(22,26,28,27)(23,31,33,32)(24,36,38,37)(25,40,42,41)(29,34,44,47)(30,39,45,46)(35,49,51,50)(43,52,53,48)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12051);