Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| 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 | |
1270 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)(10,11,15,19,16,21,14,20)(12,22,24,17,26,33,25,32)(13,27,29,18,30,35,28,36)(23,31,38,37,40,34,39,41);(10,12)(11,17)(13,23)(14,25)(15,24)(16,26)(18,34)(19,33)(20,22)(21,32)(27,37)(28,39)(29,38)(30,40)(31,36)(35,41);(7,9)(10,13,16,30)(11,18,21,36)(12,23,26,40)(14,29,15,28)(17,31,32,34)(19,27,20,35)(22,37,33,41)(24,39,25,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3993);