Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,19,31,40,14,32,17,23,13,20,16,22,33,21)(12,24,27,38,34,35,30,18,29,37,25,41,28,36,26,39);(6,7,8,9)(10,12)(11,18)(13,30)(14,25)(15,26)(16,34)(17,29)(19,35)(20,36)(21,37)(22,41)(23,39)(24,32)(27,33)(28,31)(38,40);(11,19)(13,14)(15,33)(16,31)(18,35)(20,23)(21,40)(22,32)(24,41)(25,30)(26,27)(28,34)(36,39)(37,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6007);