Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,21,16,19,14,20)(12,22,25,32,26,17,24,33)(13,18,28,36,29,35,30,37)(23,38,27,41,39,31,40,34);(10,12,16,26)(11,17,19,22)(13,27,29,40)(14,25,15,24)(18,34,35,41)(20,33,21,32)(23,30,39,28)(31,37,38,36);(7,9)(10,13)(11,18)(12,23)(14,30)(15,28)(16,29)(17,31)(19,35)(20,37)(21,36)(22,38)(24,40)(25,27)(26,39)(32,41)(33,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4003);