Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11130 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| 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 | |
862 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)(12,21)(13,18)(14,30)(15,19)(16,20)(17,23)(22,33)(24,38)(25,32)(26,41)(27,36)(28,34)(29,35)(31,39)(37,40);(10,12,14,23,16,24,15,25)(11,17,19,21,20,32,30,38)(13,26,27,22,29,39,28,40)(18,33,34,41,35,37,36,31);(10,13,16,29)(11,18,20,35)(12,22,24,40)(14,28,15,27)(17,31,32,41)(19,36,30,34)(21,37,38,33)(23,26,25,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4118);