Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C32 · Schur multiplier (H2) = C1 |
| direct product | C5×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C32 | D4 |
| commutator | C4 | C2×C5×C32 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C2×C5 | C4⋊C32 |
| prime radical | C22×C5 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C32 ◁ C2×C5×C32 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,10,17,11,18,29,42,12,19,30,43,33,45,56,64,13,16,31,39,34,40,57,61,25,41,51,62,52,63,67,69,14,20,32,44,35,46,58,65,22,15,36,37,49,38,59,60,9,26,27,53,28,54,55,68,8,21,23,47,24,48,50,66);(6,8,9,22,14,25,13,12)(7,15,16,21,20,19,26,41)(10,23,27,36,32,51,31,30)(11,24,28,49,35,52,34,33)(17,37,39,47,44,43,53,62)(18,38,40,48,46,45,54,63)(29,50,55,59,58,67,57,56)(42,60,61,66,65,64,68,69)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 532);