Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33608 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,21)(12,22,27,47)(13,18,28,39)(14,31,33,59)(15,34,32,20)(17,35,38,25)(19,42,43,23)(24,44,50,66)(26,54,53,46)(29,57,55,72)(30,58,56,41)(36,60,62,51)(37,48,45,61)(40,65,64,49)(52,71,70,68)(63,69,67,73);(10,12,23,48,15,26,33,35,16,27,42,61,32,53,14,25)(11,17,31,46,20,37,43,47,21,38,59,54,34,45,19,22)(13,24,49,69,30,52,55,60,28,50,65,73,56,70,29,51)(18,36,57,68,41,63,64,66,39,62,72,71,58,67,40,44);(7,9)(10,13)(11,18)(12,24)(14,29)(15,30)(16,28)(17,36)(19,40)(20,41)(21,39)(22,44)(23,49)(25,51)(26,52)(27,50)(31,57)(32,56)(33,55)(34,58)(35,60)(37,63)(38,62)(42,65)(43,64)(45,67)(46,68)(47,66)(48,69)(53,70)(54,71)(59,72)(61,73);(13,28)(18,39)(24,50)(29,55)(30,56)(36,62)(40,64)(41,58)(44,66)(49,65)(51,60)(52,70)(57,72)(63,67)(68,71)(69,73)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26596);