Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
1154 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);(3,4,7,13,8,14,22,31,9,11,23,27,6,12,20,29)(5,15,17,32,16,25,33,30,18,28,21,34,19,10,24,26);(3,5)(4,10)(6,19)(7,21)(8,16)(9,18)(11,25)(12,28)(13,30)(14,15)(17,23)(20,33)(22,24)(26,27)(29,32)(31,34);(4,11)(10,25)(12,14)(13,27)(15,28)(26,30)(29,31)(32,34)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26327);