Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#1476 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C23×C5 | C2×C22⋊C4 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ C22×C42×C5 ◁ G | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,5,7,6,8,9,10)(11,12,15,18,16,13,14,17);(11,13)(12,16)(14,18)(15,17);(11,14,16,15)(12,17,13,18)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10421);