Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#62637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 69 points:
(1,2,3,4,5)(6,7)(8,18)(9,14)(10,26)(11,31)(12,15)(13,20)(16,29)(17,28)(19,43)(21,50)(22,44)(23,36)(24,53)(25,56)(27,41)(30,42)(32,40)(33,46)(34,49)(35,58)(37,55)(38,52)(39,48)(45,65)(47,68)(51,64)(54,63)(57,62)(59,66)(60,69)(61,67);(6,8,12,22)(7,13,15,34)(9,19,23,45)(10,27,30,58)(11,32,28,21)(14,33,36,59)(16,40,39,50)(17,41,31,35)(18,42,44,26)(20,48,49,29)(24,54,51,69)(25,57,52,47)(37,62,61,68)(38,63,56,60)(43,64,65,53)(46,67,66,55);(6,9)(7,14)(8,19)(10,24)(11,25)(12,23)(13,33)(15,36)(16,37)(17,38)(18,43)(20,46)(21,47)(22,45)(26,53)(27,54)(28,52)(29,55)(30,51)(31,56)(32,57)(34,59)(35,60)(39,61)(40,62)(41,63)(42,64)(44,65)(48,67)(49,66)(50,68)(58,69);(9,23)(14,36)(19,45)(24,51)(25,52)(33,59)(37,61)(38,56)(43,65)(46,66)(47,57)(53,64)(54,69)(55,67)(60,63)(62,68);(7,15)(13,34)(14,36)(16,39)(17,31)(18,44)(26,42)(33,59)(35,41)(37,61)(38,56)(40,50)(43,65)(53,64)(60,63)(62,68)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55625);