Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C64 · Schur multiplier (H2) = C22 |
| direct product | C5×C22⋊C64 |
| unambiguous semidirect product | C22⋊(C5×C64) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C32 | C22 |
| commutator | C2 | C2×C5×C64 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C22×C5 | C2×C32 |
| prime radical | C23×C5 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C32 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,17,13,18,22,32,14,19,23,33,26,36,42,52,15,20,24,34,27,37,43,53,29,39,45,55,48,58,62,67,16,21,25,35,28,38,44,54,30,40,46,56,49,59,63,68,31,41,47,57,50,60,64,69,51,61,65,70,66,71,72,73);(7,9)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 500);