Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2163 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,8)(6,7)(9,10)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,16)(17,22)(18,23)(24,26)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26404);