Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #396 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
822 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)(10,11,15,19,16,20,14,31)(12,21,25,33,24,37,23,17)(13,26,29,35,30,18,28,34)(22,38,40,41,27,36,39,32);(6,7,8,9)(10,12,14,23,16,24,15,25)(11,17,19,21,20,33,31,37)(13,27,28,40,30,22,29,39)(18,32,34,38,26,41,35,36);(10,13,16,30)(11,18,20,26)(12,22,24,27)(14,29,15,28)(17,32,33,41)(19,35,31,34)(21,36,37,38)(23,40,25,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4128);