Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3167 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C2×C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,8)(6,7)(9,10)(12,15)(13,17)(16,18);(4,7)(8,10);(11,12,13,15,14,16,17,18);(3,5,6,9)(4,8,7,10)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53657);