Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11074 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
866 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)(7,9)(10,11,15,21,16,20,14,19)(12,17,23,31,24,32,29,36)(13,25,27,18,28,35,26,34)(22,37,38,33,39,41,40,30);(6,7,8,9)(10,12)(11,17)(13,22)(14,29)(15,23)(16,24)(18,33)(19,36)(20,32)(21,31)(25,37)(26,40)(27,38)(28,39)(30,34)(35,41);(10,13,16,28)(11,18,20,34)(12,22,24,39)(14,27,15,26)(17,30,32,33)(19,35,21,25)(23,40,29,38)(31,41,36,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4062);