Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
1078 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)(8,20)(9,25)(10,29)(11,19)(12,18)(13,16)(14,22)(15,27)(17,32)(21,48)(23,54)(24,47)(26,57)(28,44)(30,45)(31,41)(33,43)(34,39)(35,42)(36,63)(37,53)(38,61)(40,60)(46,50)(49,55)(51,68)(52,66)(56,59)(58,64)(62,67)(65,69);(6,8,13,24)(7,14,16,37)(9,21,28,52)(10,30,33,61)(11,23,12,35)(15,36,39,64)(17,42,41,54)(18,38,19,45)(20,43,47,29)(22,31,53,32)(25,55,44,65)(26,58,60,63)(27,50,34,51)(40,66,57,48)(46,67,68,56)(49,62,69,59);(6,9)(7,15)(8,21)(10,26)(11,34)(12,27)(13,28)(14,36)(16,39)(17,40)(18,44)(19,25)(20,46)(22,49)(23,51)(24,52)(29,56)(30,58)(31,62)(32,59)(33,60)(35,50)(37,64)(38,65)(41,57)(42,66)(43,67)(45,55)(47,68)(48,54)(53,69)(61,63);(7,16)(14,37)(15,39)(17,41)(18,19)(20,47)(25,44)(29,43)(36,64)(38,45)(40,57)(42,54)(46,68)(48,66)(55,65)(56,67)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26528);