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×#33806 |
| 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 | |
462 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)(24,34)(25,33)(26,29)(27,40)(28,31)(30,37)(32,42)(35,51)(36,50)(38,45)(39,47)(41,49)(43,48)(44,52)(46,53);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(23,29)(30,44)(31,33)(32,45)(34,50)(40,47)(46,49)(48,51);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,24,26,36)(23,30,29,44)(25,35,28,43)(27,41,39,53)(31,46,33,49)(32,48,45,51)(34,47,50,40)(37,38,52,42);(7,13)(11,14)(12,21)(19,20)(22,25,26,28)(23,31,29,33)(24,35,36,43)(27,38,39,42)(30,46,44,49)(32,47,45,40)(34,48,50,51)(37,41,52,53)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26794);