Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C5 |
| prime radical | C2×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24)(25,26,27,28,29);(1,3,4,7)(2,5,6,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(2,6)(5,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 989);