Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25189 |
| 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 | |
1000 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)(10,11,16,20,17,21,15,31)(12,22,24,18,13,25,27,34)(14,28,29,19,30,35,32,36)(23,33,39,40,26,41,38,37);(6,7)(8,9)(10,12,16,24,17,13,15,27)(11,18,20,25,21,34,31,22)(14,23,29,39,30,26,32,38)(19,33,35,40,36,41,28,37);(7,9)(10,13)(11,18)(12,17)(14,26)(15,27)(16,24)(19,33)(20,22)(21,34)(23,30)(25,31)(28,40)(29,39)(32,38)(35,37)(36,41);(10,14,17,30)(11,19,21,36)(12,23,13,26)(15,32,16,29)(18,33,34,41)(20,35,31,28)(22,37,25,40)(24,39,27,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18177);