Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1328 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)(12,20)(13,27)(14,31)(15,22)(16,21)(17,23)(18,25)(19,29)(24,38)(26,34)(28,41)(30,37)(32,36)(33,40)(35,39);(6,7)(8,9)(10,12,15,25,17,26,16,14)(11,18,21,20,23,31,22,34)(13,24,29,39,30,40,32,28)(19,33,27,35,36,38,37,41);(7,9)(10,13)(11,19)(12,24)(14,28)(15,29)(16,32)(17,30)(18,33)(20,35)(21,27)(22,37)(23,36)(25,39)(26,40)(31,38)(34,41);(10,14,16,26,17,25,15,12)(11,20,22,18,23,34,21,31)(13,28,32,40,30,39,29,24)(19,35,37,33,36,41,27,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17268);