Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12747 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1786 |
| prime radical | C5×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
886 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)(8,19)(9,24,27,42)(10,28)(11,17)(12,34)(13,18)(14,21)(15,26,41,33)(16,31)(20,48,51,68)(22,46)(23,47)(25,40,58,66)(29,37)(30,44)(32,60)(35,43)(36,63,64,56)(38,61)(39,53)(45,50,67,52)(49,54,69,65)(55,62,59,57);(6,8)(7,14)(9,20)(10,29)(11,22)(12,35)(13,23)(15,36)(16,43)(17,37)(18,39)(19,28)(21,31)(24,54)(25,56)(26,50)(27,51)(30,53)(32,61)(33,52)(34,38)(40,48)(41,64)(42,65)(44,46)(45,59)(47,60)(49,57)(55,67)(58,63)(62,69)(66,68);(6,9)(7,15)(8,20)(10,25)(11,33)(12,26)(13,27)(14,36)(16,40)(17,42)(18,41)(19,45)(21,49)(22,52)(23,51)(24,34)(28,59)(29,56)(30,62)(31,57)(32,58)(35,50)(37,65)(38,54)(39,64)(43,48)(44,66)(46,68)(47,67)(53,69)(55,60)(61,63)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5735);