Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1879 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
456 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)(6,7,11,17,36,39,34,16,13,19,33,50,12,18,10,32)(8,20,23,14,37,41,22,51,25,54,61,43,24,48,35,42)(9,26,29,47,38,49,27,46,31,44,64,15,30,45,28,52)(21,40,57,53,62,69,56,65,59,68,60,63,58,67,55,66);(6,8,12,37,13,25,36,24)(7,14,39,42,19,43,18,51)(9,21,38,58,31,59,30,62)(10,23,33,22,34,61,11,35)(15,40,46,69,47,68,52,67)(16,48,50,54,32,41,17,20)(26,63,45,66,44,53,49,65)(27,60,64,56,28,57,29,55);(6,9)(7,15)(8,21)(10,29)(11,28)(12,38)(13,31)(14,40)(16,49)(17,44)(18,52)(19,47)(20,53)(22,60)(23,55)(24,62)(25,59)(26,50)(27,33)(30,36)(32,45)(34,64)(35,57)(37,58)(39,46)(41,66)(42,69)(43,68)(48,65)(51,67)(54,63)(56,61)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6019);