Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | #389 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,8,12,9,13,7,23)(5,14,16,10,18,25,17,26)(6,19,21,27,22,33,20,11)(15,24,29,32,31,34,30,28);(3,5,8,16,9,18,7,17)(4,10,12,25,13,26,23,14)(6,15,21,29,22,31,20,30)(11,24,19,32,27,34,33,28);(3,6,7,20,9,22,8,21)(4,11,12,19,13,27,23,33)(5,15,17,30,18,31,16,29)(10,24,25,32,26,34,14,28)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14171);