Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21181 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
874 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)(6,7,9,8)(10,11)(12,22)(13,27)(14,19)(15,20)(16,21)(17,25)(18,29)(23,37)(24,32)(26,35)(28,34)(30,33)(31,40)(36,39)(38,41);(6,8,9,7)(10,12)(11,17)(13,23)(14,24)(15,25)(16,26)(18,31)(19,22)(20,35)(21,32)(27,36)(28,38)(29,39)(30,40)(33,41)(34,37);(10,13,15,29,16,30,14,28)(11,18,19,27,21,34,20,33)(12,23,25,39,26,40,24,38)(17,31,22,36,32,37,35,41);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,28,30,29)(17,22,32,35)(18,27,34,33)(23,38,40,39)(31,36,37,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14169);