Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(21,27)(24,30)(28,33)(31,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10363);