Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10990 |
| 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 | |
1278 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)(10,11,15,19,16,20,14,29)(12,21,23,17,24,33,30,32)(13,25,27,36,28,18,26,35)(22,38,39,31,40,37,41,34);(10,12)(11,17)(13,22)(14,30)(15,23)(16,24)(18,34)(19,33)(20,32)(21,29)(25,31)(26,41)(27,39)(28,40)(35,38)(36,37);(7,9)(10,13,16,28)(11,18,20,25)(12,22,24,40)(14,27,15,26)(17,31,32,34)(19,36,29,35)(21,37,33,38)(23,41,30,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3978);