Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C5 |
| prime radical | C24×C5 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4,5,6,7);(8,9)(10,14)(11,12)(13,15)(16,17)(18,20)(19,21)(22,23);(9,12)(13,14)(16,18,19,22)(17,20,21,23);(1,2)(8,10,11,15)(9,13,12,14)(17,21)(20,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1530);