Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#2108 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C16 | C22 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C16 ◁ G | |
854 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);(7,8,10,14,11,15,20,26,12,16,21,27,22,28,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(3,4,5,6)(7,9)(8,13)(10,18)(11,19)(12,17)(14,24)(15,25)(16,23)(20,31)(21,29)(22,30)(26,35)(27,33)(28,34)(32,37)(36,38);(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26311);