Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
436 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)(8,18,23,41)(9,24)(10,29,19,15)(11,32)(12,17)(13,21)(14,27)(16,25)(20,44,48,67)(22,50,45,40)(26,55,42,52)(28,53)(30,35)(31,38,43,59)(33,62)(34,60)(36,57,58,66)(37,54)(39,64,65,47)(46,51)(49,61)(56,68)(63,69);(6,8,19,21,12,23,10,30)(7,13,29,41,17,35,15,18)(9,20,42,46,28,48,26,56)(11,22,43,49,25,45,31,60)(14,33,58,65,37,63,36,39)(16,34,59,50,32,61,38,40)(24,51,55,67,53,68,52,44)(27,47,66,62,54,64,57,69);(6,9,25,54,12,28,11,27)(7,14,32,53,17,37,16,24)(8,20,45,64,23,48,22,47)(10,26,43,66,19,42,31,57)(13,33,61,68,35,63,34,51)(15,36,59,55,29,58,38,52)(18,39,50,67,41,65,40,44)(21,46,60,69,30,56,49,62)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6218);