Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1682 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)(10,11)(12,17)(13,25)(14,19)(15,30)(16,21)(18,28)(20,23)(22,35)(24,34)(26,41)(27,37)(29,33)(31,36)(32,39)(38,40);(6,7)(8,9)(10,12,16,24)(11,17,21,34)(13,22,27,38)(14,15,29,23)(18,32,36,41)(19,30,33,20)(25,40,37,35)(26,31,39,28);(7,9)(10,13)(11,18)(12,22)(14,26)(15,31)(16,27)(17,32)(19,35)(20,37)(21,36)(23,28)(24,38)(25,30)(29,39)(33,40)(34,41);(10,14,16,29)(11,19,21,33)(12,23,24,15)(13,26,27,39)(17,20,34,30)(18,35,36,40)(22,28,38,31)(25,32,37,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16905);