Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22285 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2328 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
772 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)(8,19)(9,25,29,39)(10,30)(11,17)(12,18)(13,22)(14,28,40,54)(15,32)(16,31)(20,45,48,34)(21,49)(23,36)(24,43)(26,37,47,62)(27,38,57,63)(33,41)(35,52)(42,59)(44,50)(46,66,68,61)(51,64)(53,69,58,56)(55,65,67,60);(6,8,11,22,12,24,31,23)(7,13,16,19,18,36,17,43)(9,26,54,20,29,47,28,48)(10,21,32,50,33,52,59,51)(14,37,25,34,40,62,39,45)(15,35,30,44,42,49,41,64)(27,55,69,46,57,67,56,68)(38,61,53,60,63,66,58,65);(6,9,12,29)(7,14)(8,20)(10,27,33,57)(11,28,31,54)(13,34,36,45)(15,38)(16,39)(17,25)(18,40)(19,37,43,62)(21,46)(22,26)(23,47)(24,48)(30,58)(32,56,59,69)(35,60,49,65)(41,53)(42,63)(44,61,64,66)(50,55)(51,67)(52,68);(6,10,31,59,12,33,11,32)(7,15,17,41,18,42,16,30)(8,21,23,51,24,52,22,50)(9,27,54,69,29,57,28,56)(13,35,43,64,36,49,19,44)(14,38,25,53,40,63,39,58)(20,46,47,67,48,68,26,55)(34,60,62,66,45,65,37,61)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15273);