Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20955 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1710 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)(7,9)(10,11,15,19,16,21,14,20)(12,22,24,17,26,32,25,33)(13,27,28,34,30,18,29,35)(23,36,39,40,38,41,37,31);(6,7,8,9)(10,12)(11,17)(13,23)(14,24)(15,25)(16,26)(18,31)(19,22)(20,32)(21,33)(27,40)(28,37)(29,39)(30,38)(34,36)(35,41);(10,13)(11,18)(12,23)(14,28)(15,29)(16,30)(17,31)(19,34)(20,35)(21,27)(22,36)(24,37)(25,39)(26,38)(32,41)(33,40);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,28,30,29)(17,22,33,32)(18,34,27,35)(23,37,38,39)(31,36,40,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13943);