Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23438 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1922 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)(12,22)(13,27)(14,19)(15,20)(16,21)(17,25)(18,30)(23,38)(24,34)(26,33)(28,35)(29,37)(31,36)(32,40)(39,41);(7,9)(10,12)(11,17)(13,28)(14,24)(15,25)(16,26)(18,35)(19,22)(20,33)(21,34)(23,29)(27,38)(30,40)(31,39)(32,36)(37,41);(10,13)(11,18)(12,23)(14,29)(15,30)(16,31)(17,32)(19,27)(20,36)(21,37)(22,35)(24,39)(25,28)(26,40)(33,41)(34,38);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,29,31,30)(17,22,34,33)(18,27,37,36)(23,39,40,28)(32,35,38,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16426);