Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22 |
| commutator | C2 | C22×C5×C8 |
| Frattini, cosocle | C8 | C23×C5 |
| socle | C22×C5 | C2×C8 |
| prime radical | C23×C5 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18);(4,12)(6,14)(8,16)(10,18)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1004);