Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33584 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
722 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)(7,9)(10,11)(12,22)(13,21)(14,17)(15,28)(16,19)(18,25)(20,30)(23,39)(24,38)(26,33)(27,35)(29,37)(31,36)(32,40)(34,41);(6,7,8,9)(11,17)(18,32)(19,21)(20,33)(22,38)(28,35)(34,37)(36,39);(10,12)(11,18)(13,23)(14,24)(15,29)(16,31)(17,32)(19,34)(20,36)(21,37)(22,28)(25,30)(26,40)(27,41)(33,39)(35,38);(10,13,14,16)(11,19,17,21)(12,23,24,31)(15,26,27,30)(18,34,32,37)(20,35,33,28)(22,36,38,39)(25,29,40,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26572);