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) = C24 |
| direct product | C2×C5×D4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | D4 |
| commutator | C4 | C22×C5×C16 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C23×C5 | C22⋊C8 |
| prime radical | C22×C5×D4 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,7)(6,9)(8,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(4,7)(5,10)(8,9)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4569);