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×#24282 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1308 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)(10,11,17,23)(12,24,14,32)(13,28,31,37)(15,22,16,21)(18,27,20,26)(19,33,36,30)(25,35,29,34)(38,40,41,39);(6,7)(8,9)(10,12,16,27,17,14,15,26)(11,18,22,32,23,20,21,24)(13,25,33,40,31,29,30,39)(19,34,37,41,36,35,28,38);(7,9)(10,13)(11,19)(12,25)(14,29)(15,30)(16,33)(17,31)(18,34)(20,35)(21,28)(22,37)(23,36)(24,38)(26,39)(27,40)(32,41);(10,14,16,26,17,12,15,27)(11,20,22,24,23,18,21,32)(13,29,33,39,31,25,30,40)(19,35,37,38,36,34,28,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17270);