Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| 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 | |
1238 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)(10,11,15,20,16,19,14,31)(12,21,24,36,25,17,23,33)(13,26,30,37,28,18,29,35)(22,32,40,34,39,38,27,41);(10,12,16,25)(11,17,19,21)(13,27,28,40)(14,24,15,23)(18,34,26,41)(20,36,31,33)(22,30,39,29)(32,37,38,35);(7,9)(10,13)(11,18)(12,22)(14,30)(15,29)(16,28)(17,32)(19,26)(20,37)(21,38)(23,40)(24,27)(25,39)(31,35)(33,41)(34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4023);