Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1579 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C2×D4 |
| commutator | C4 | C22×C3×C5×C8 |
| Frattini, cosocle | C42 | C23×C3×C5 |
| socle, prime radical | C22×C3×C5 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C3×C4×C5×C8 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 1920 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3)(4,5,6,7,8)(9,10,12,15,13,11,14,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(9,11,12,16,13,10,14,15)(18,23)(22,31)(24,28)(25,32)660 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 311);