Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33803 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
456 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)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,28,32)(24,33,34,50)(25,30,27,42)(26,38,41,52)(29,44,45,36)(31,37,49,40)(35,51,43,48)(39,46,53,47);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(24,34)(29,45)(33,50)(35,43)(36,44)(39,53)(46,47)(48,51);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,24)(23,29)(25,35)(26,39)(27,43)(28,34)(30,46)(31,48)(32,45)(33,38)(36,40)(37,44)(41,53)(42,47)(49,51)(50,52);(7,13)(11,14)(12,21)(19,20)(22,25,28,27)(23,30,32,42)(24,35,34,43)(26,37,41,40)(29,46,45,47)(31,38,49,52)(33,51,50,48)(36,39,44,53)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26791);