Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #380 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
320 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)(6,7,10,13,9,11,17,19)(8,14,16,21,15,12,18,20)(22,23)(24,33)(25,38,40,45)(26,41)(27,31)(28,32)(29,35)(30,39,46,42)(34,43,50,53)(36,44)(37,48)(47,49,52,51);(7,11)(12,14)(13,19)(20,21)(22,24,26,35,28,37,27,36)(23,29,31,33,32,44,41,48)(25,34,39,49,40,50,42,51)(30,43,45,47,46,53,38,52);(6,8,9,15)(7,12,11,14)(10,18,17,16)(13,21,19,20)(22,25)(23,30)(24,34)(26,39)(27,42)(28,40)(29,43)(31,45)(32,46)(33,47)(35,49)(36,51)(37,50)(38,41)(44,53)(48,52)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1623);