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×#24279 |
| 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)(12,24)(13,28)(14,18)(15,22)(16,21)(17,23)(19,30)(20,27)(25,39)(26,34)(29,38)(31,37)(32,36)(33,40)(35,41);(6,7)(8,9)(10,12,16,26,17,27,15,14)(11,18,22,20,23,34,21,24)(13,25,32,41,31,40,30,29)(19,33,37,35,36,39,28,38);(7,9)(10,13)(11,19)(12,25)(14,29)(15,30)(16,32)(17,31)(18,33)(20,35)(21,28)(22,37)(23,36)(24,38)(26,41)(27,40)(34,39);(10,14,15,27,17,26,16,12)(11,20,21,18,23,24,22,34)(13,29,30,40,31,41,32,25)(19,35,28,33,36,38,37,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17267);