Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #102 |
| commutator | C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| prime radical | C22×C5 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,5,9,6,11,21,10)(3,12,14,25,15,29,20,28)(4,16,18,8,19,26,32,27)(7,17,22,13,24,30,23,31)(33,34,35,36,37);(1,3,6,15)(2,7,11,24)(4,17,19,30)(5,20,21,14)(8,25,27,28)(9,23,10,22)(12,16,29,26)(13,18,31,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 167);