Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23530 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1350 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)(23,32)(24,34)(26,33)(28,40)(29,36)(31,35)(37,38)(39,41);(10,12)(11,17)(13,28)(14,24)(15,25)(16,26)(18,32)(19,22)(20,33)(21,34)(23,31)(27,37)(29,39)(30,38)(35,41)(36,40);(7,9)(10,13,14,29,16,31,15,30)(11,18,20,35,21,36,19,27)(12,23,25,39,26,28,24,38)(17,32,22,37,34,40,33,41);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,29,31,30)(17,22,34,33)(18,27,36,35)(23,38,28,39)(32,37,40,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16518);