Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9286 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(7,11,13,16)(8,14,15,17)(22,23,27,31,28,32,26,42)(24,33,37,44,36,48,35,29)(25,38,40,46,41,30,39,45)(34,47,51,43,50,53,49,52);(6,7,9,16,10,13,18,11)(8,12,17,20,15,21,14,19)(22,24,28,36)(23,29)(25,34,41,50)(26,35,27,37)(30,43)(31,33)(32,44)(38,52)(39,49,40,51)(42,48)(45,53)(46,47);(6,8)(7,12)(9,17)(10,15)(11,19)(13,21)(14,18)(16,20)(22,25,28,41)(23,30,32,38)(24,34,36,50)(26,40,27,39)(29,43,44,52)(31,46,42,45)(33,47,48,53)(35,51,37,49)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2274);