Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C8 | D4 |
| commutator | C4 | C22×C5×C16 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C4⋊C8 |
| prime radical | C23×C5 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,19)(11,14)(12,17)(16,20)(18,21)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37);(6,8,10,17,11,18,9,16)(7,12,13,15,14,20,19,21);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5000);