Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1874 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
472 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)(6,7,12,18,37,40,35,16,13,19,34,53,11,17,10,32)(8,20,24,14,36,43,22,51,25,55,64,44,23,49,38,42)(9,26,29,45,28,48,33,52,31,47,39,50,30,15,27,46)(21,56,59,65,58,66,57,41,61,68,63,67,60,54,62,69);(6,8,11,36,13,25,37,23)(7,14,40,42,19,44,17,51)(9,21,30,58,31,61,28,60)(10,24,34,22,35,64,12,38)(15,41,26,65,48,69,47,67)(16,49,53,55,32,43,18,20)(27,59,39,57,33,63,29,62)(45,56,52,54,50,68,46,66);(6,9)(7,15)(8,21)(10,33)(11,30)(12,39)(13,31)(14,41)(16,50)(17,47)(18,52)(19,48)(20,54)(22,62)(23,60)(24,63)(25,61)(26,40)(27,35)(28,37)(29,34)(32,45)(36,58)(38,57)(42,65)(43,56)(44,69)(46,53)(49,68)(51,67)(55,66)(59,64)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5994);