Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33538 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5)(6,7)(8,20)(9,25)(10,16)(11,31)(12,18)(13,19)(14,23)(15,28)(17,33)(21,49)(22,47)(24,48)(26,41)(27,58)(29,45)(30,44)(32,37)(34,43)(35,42)(36,63)(38,62)(39,54)(40,61)(46,52)(50,68)(51,55)(53,66)(56,60)(57,67)(59,65)(64,69);(6,8)(7,14)(9,21)(10,22)(11,32)(12,35)(13,24)(15,36)(16,37)(17,42)(18,38)(19,39)(20,31)(23,33)(25,55)(26,50)(27,59)(28,52)(29,53)(30,54)(34,62)(40,66)(41,65)(43,48)(44,47)(45,64)(46,56)(49,58)(51,60)(57,69)(61,63)(67,68);(6,9)(7,15)(8,21)(10,26)(11,27)(12,28)(13,29)(14,36)(16,25)(17,40)(18,45)(19,41)(20,46)(22,50)(23,51)(24,53)(30,57)(31,56)(32,59)(33,60)(34,61)(35,52)(37,55)(38,64)(39,65)(42,66)(43,67)(44,58)(47,49)(48,68)(54,69)(62,63);(6,10,13,12)(7,16,19,18)(8,22,24,35)(9,26,29,28)(11,30,34,33)(14,37,39,38)(15,25,41,45)(17,31,44,43)(20,47,48,42)(21,50,53,52)(23,32,54,62)(27,57,61,60)(36,55,65,64)(40,56,58,67)(46,49,68,66)(51,59,69,63)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26526);