Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C22×#107 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
622 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)(6,7,11,17,13,18,12,33)(8,19,23,44,25,45,24,53)(9,15,28,40,26,38,29,57)(10,30,31,41,32,58,20,16)(14,34,36,52,37,46,43,22)(21,42,50,65,48,63,51,68)(27,55,56,62,54,69,47,39)(35,59,61,67,60,66,64,49);(6,8,20,46,12,24,32,52,13,25,31,34,11,23,10,22)(7,14,30,44,17,36,41,45,18,37,58,53,33,43,16,19)(9,21,47,66,29,51,54,67,26,48,56,59,28,50,27,49)(15,35,55,65,40,61,62,63,38,60,69,68,57,64,39,42);(6,9)(7,15)(8,21)(10,27)(11,28)(12,29)(13,26)(14,35)(16,39)(17,40)(18,38)(19,42)(20,47)(22,49)(23,50)(24,51)(25,48)(30,55)(31,56)(32,54)(33,57)(34,59)(36,61)(37,60)(41,62)(43,64)(44,65)(45,63)(46,66)(52,67)(53,68)(58,69);(9,26)(15,38)(21,48)(27,54)(28,29)(35,60)(39,62)(40,57)(42,63)(47,56)(49,67)(50,51)(55,69)(59,66)(61,64)(65,68)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26402);