Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1270 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)(10,11)(12,20)(13,24,26,17)(14,27)(15,19)(16,22)(18,29)(21,35,37,34)(23,33)(25,32,41,39)(28,31)(30,36,38,40);(6,7)(8,9)(10,12,15,23)(11,16,19,31)(13,21,26,37)(14,28,29,22)(17,30,24,38)(18,33,27,20)(25,40,41,36)(32,35,39,34);(7,9)(10,13)(11,17)(12,21)(14,25)(15,26)(16,30)(18,32)(19,24)(20,34)(22,36)(23,37)(27,39)(28,40)(29,41)(31,38)(33,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3973);