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) = C23 |
| direct product | C5×#12945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,23)(13,17)(14,32)(15,33)(16,22)(18,25)(19,28)(20,31)(21,30)(24,37)(26,36)(27,39)(29,38)(34,35)(40,41);(6,8,9,7)(11,17)(13,28)(14,31)(15,30)(18,35)(19,21)(20,22)(23,37)(24,25)(26,34)(29,41)(32,33)(36,38)(39,40);(10,12)(11,18)(13,29)(14,24)(15,34)(16,27)(17,35)(19,39)(20,36)(21,40)(22,38)(23,32)(25,31)(26,30)(28,41)(33,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5933);