Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,21,16,20,14,19)(12,17,25,32,24,33,23,31)(13,26,27,18,28,34,29,35)(22,36,37,30,39,40,38,41);(10,12,14,25,16,24,15,23)(11,17,21,31,20,33,19,32)(13,22,27,38,28,39,29,37)(18,30,26,40,35,41,34,36);(7,9)(10,13)(11,18)(12,22)(14,27)(15,29)(16,28)(17,30)(19,34)(20,35)(21,26)(23,37)(24,39)(25,38)(31,40)(32,36)(33,41);(10,14,16,15)(11,19,20,21)(12,23,24,25)(13,27,28,29)(17,31,33,32)(18,34,35,26)(22,37,39,38)(30,40,41,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13994);