Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C5×#1764 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C4×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ C23×C5×C8 ◁ G | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4)(5,6)(7,8,11,15,12,16,10,13)(9,14,18,21,19,22,17,20);(4,6)(8,13,16,15)(14,20,22,21);(7,9,11,18,12,19,10,17)(8,14,15,21,16,22,13,20)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13406);