Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1294 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)(8,18,22,13)(9,23)(10,28,30,15)(11,16)(12,17)(14,26)(19,44,47,34)(20,35)(21,42,32,36)(24,51,50,37)(25,52,54,38)(27,43)(29,56,59,40)(31,41)(33,39)(45,48)(46,66)(49,61)(53,69)(55,65,58,63)(57,64)(60,67)(62,68);(6,8,11,22)(7,13,16,18)(9,24,26,50)(10,20,30,48)(12,32,31,21)(14,37,23,51)(15,35,28,45)(17,42,41,36)(19,33,47,27)(25,53,54,68)(29,57,59,49)(34,43,44,39)(38,62,52,69)(40,64,56,61)(46,58,67,55)(60,65,66,63);(6,9)(7,14)(8,19)(10,25)(11,26)(12,33)(13,34)(15,38)(16,23)(17,43)(18,44)(20,46)(21,50)(22,47)(24,32)(27,31)(28,52)(29,58)(30,54)(35,60)(36,51)(37,42)(39,41)(40,65)(45,66)(48,67)(49,68)(53,57)(55,59)(56,63)(61,69)(62,64);(6,10)(7,15)(8,20)(9,25)(11,30)(12,29)(13,35)(14,38)(16,28)(17,40)(18,45)(19,46)(21,49)(22,48)(23,52)(24,53)(26,54)(27,55)(31,59)(32,57)(33,58)(34,60)(36,61)(37,62)(39,63)(41,56)(42,64)(43,65)(44,66)(47,67)(50,68)(51,69)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17399);