Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1290 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)(12,18)(13,26)(14,29)(15,22)(16,21)(17,24)(19,30)(20,27)(23,38)(25,33)(28,32)(31,37)(34,39)(35,40)(36,41);(10,12,15,24,16,13,27,25)(11,17,20,18,21,33,22,26)(14,23,30,39,31,28,41,40)(19,32,29,34,36,38,37,35);(7,9)(10,13)(11,18)(12,16)(14,23)(15,25)(17,22)(19,35)(20,33)(21,26)(24,27)(28,31)(29,32)(30,39)(34,36)(37,38)(40,41);(10,14,27,41,16,31,15,30)(11,19,22,37,21,36,20,29)(12,23,25,40,13,28,24,39)(17,32,26,35,33,38,18,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17281);