Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(39,40,41,42,43);(3,4)(5,6)(7,8,12,16)(9,17,20,32)(10,21,24,14)(11,25,27,15)(13,26,29,19)(18,34,22,33)(23,31,38,36)(28,35,30,37);(7,9,12,20)(8,13,16,29)(10,22,24,18)(11,26,27,19)(14,30,21,28)(15,32,25,17)(23,35,38,37)(31,33,36,34);(4,6)(7,10,11,23,12,24,27,38)(8,14,15,31,16,21,25,36)(9,18,19,35,20,22,26,37)(13,28,17,33,29,30,32,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16507);