Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1994 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)(12,16)(13,17)(14,18)(15,19)(20,27)(21,28)(22,29)(23,31)(24,32)(25,33)(26,34)(30,36)(35,39)(37,40)(38,41);(10,12,15,22)(11,16,19,29)(13,20,24,36)(14,25,26,21)(17,30,32,27)(18,33,34,28)(23,37,38,35)(31,39,41,40);(7,9)(10,13)(11,17)(12,20)(14,23)(15,24)(16,27)(18,31)(19,32)(21,35)(22,36)(25,37)(26,38)(28,39)(29,30)(33,40)(34,41);(10,14,15,26)(11,18,19,34)(12,21,22,25)(13,23,24,38)(16,28,29,33)(17,31,32,41)(20,35,36,37)(27,39,30,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16434);