Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
480 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)(10,11,16,23,19,28,15,27,18,24,14,21,17,12,25,22)(13,29,32,20,36,38,31,26,34,41,30,40,33,37,35,39);(6,7,8,9)(10,12,18,28)(11,19,24,17)(13,26,34,39)(14,23,16,21)(15,27,25,22)(20,33,40,36)(29,35,41,31)(30,37,32,38);(10,13,18,34)(11,20,24,40)(12,26,28,39)(14,31,16,35)(15,30,25,32)(17,36,19,33)(21,41,23,29)(22,38,27,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6006);