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×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
516 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)(10,11,13,18,28,37,14,30,16,21,29,38,15,20,27,19)(12,22,23,39,32,34,41,36,26,40,31,35,25,33,24,17);(6,7,8,9)(10,12)(11,17)(13,24)(14,31)(15,32)(16,26)(18,33)(19,22)(20,39)(21,36)(23,27)(25,28)(29,41)(30,40)(34,38)(35,37);(11,18)(13,27)(14,29)(15,28)(17,33)(19,37)(20,30)(21,38)(22,35)(23,24)(25,32)(31,41)(34,36)(39,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6005);