Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
468 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)(24,28)(25,29)(26,39)(27,32)(30,36)(31,37)(33,40)(34,49)(35,43)(38,45)(41,46)(42,47)(44,50)(48,52)(51,53);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24,27,35)(23,28,32,43)(25,33,38,48)(26,34,37,47)(29,40,45,52)(30,41,44,51)(31,42,39,49)(36,46,50,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(25,36)(26,37)(28,40)(30,31)(32,45)(33,46)(34,47)(38,50)(39,44)(41,42)(43,52)(48,53)(49,51);(7,13)(11,14)(12,21)(19,20)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26786);