Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,20)(13,17)(14,26)(15,19)(16,22)(18,28)(21,36)(23,35)(24,33)(25,32)(27,31)(29,37)(30,40)(34,38)(39,41);(10,12)(11,16,19,30)(13,21)(14,22)(15,23)(17,29,32,41)(18,20,26,35)(24,37)(25,38)(27,39)(28,40)(31,36,33,34);(10,13,15,25)(11,17,19,32)(12,21,23,38)(14,27,28,24)(16,29,30,41)(18,33,26,31)(20,34,35,36)(22,39,40,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3729);