Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,20)(9,26)(10,30)(11,19)(12,17)(13,18)(14,22)(15,28)(16,33)(21,50)(23,43)(24,48)(25,49)(27,59)(29,46)(31,37)(32,45)(34,44)(35,42)(36,60)(38,55)(39,54)(40,65)(41,62)(47,52)(51,57)(53,67)(56,66)(58,61)(63,68)(64,69);(6,8,13,25)(7,14,18,38)(9,21,29,53)(10,31,34,54)(11,24,12,23)(15,40,42,36)(16,43,45,48)(17,39,19,37)(20,44,49,30)(22,32,55,33)(26,57,46,64)(27,60,62,65)(28,56,35,52)(41,67,59,50)(47,58,66,68)(51,63,69,61);(6,9)(7,15)(8,21)(10,27)(11,35)(12,28)(13,29)(14,36)(16,41)(17,46)(18,42)(19,26)(20,47)(22,51)(23,56)(24,52)(25,53)(30,58)(31,60)(32,63)(33,61)(34,62)(37,64)(38,40)(39,57)(43,50)(44,68)(45,59)(48,67)(49,66)(54,65)(55,69)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6299);