Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7)(8,13)(9,15)(10,14)(11,16)(12,17)(18,21)(19,20)(22,23,27,31)(24,28,34,41)(25,29,36,43)(26,37,38,30)(32,44,46,39)(33,47,48,40)(35,49,50,42)(45,51,53,52);(6,8)(7,12)(9,17)(10,19)(11,18)(13,14)(15,21)(16,20)(22,24)(23,28)(25,32)(26,33)(27,34)(29,39)(30,40)(31,41)(35,45)(36,46)(37,47)(38,48)(42,51)(43,44)(49,52)(50,53);(6,8)(7,13)(9,19)(10,17)(11,18)(12,14)(15,20)(16,21)(22,25)(23,29)(24,32)(26,35)(27,36)(28,39)(30,42)(31,43)(33,45)(34,46)(37,49)(38,50)(40,51)(41,44)(47,52)(48,53);(6,9,11,10)(7,14,16,15)(8,17,18,19)(12,13,20,21)(22,26,27,38)(23,30,31,37)(24,33,34,48)(25,35,36,50)(28,40,41,47)(29,42,43,49)(32,45,46,53)(39,51,44,52)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11458);