Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
692 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)(12,20)(13,19)(14,28)(16,17)(18,23)(21,33)(22,34)(24,37)(26,27)(29,31)(32,36)(35,41)(38,40);(10,12)(11,18)(13,21)(14,29)(15,20)(16,31)(17,22)(19,32)(23,28)(24,38)(25,33)(26,40)(27,35)(30,34)(36,37)(39,41);(7,9)(10,13,17,27)(11,19,16,26)(12,21,22,35)(14,24,28,37)(15,25,30,39)(18,32,31,40)(20,33,34,41)(23,36,29,38);(10,14)(11,15)(12,22)(13,24)(16,30)(17,28)(19,25)(20,29)(21,35)(23,34)(26,39)(27,37)(33,38)(36,41)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26821);