Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#444 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×D4 |
| commutator | C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| socle | C2×C5 | C4×D4 |
| prime radical | C5×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18,19,20,21);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16);(1,4,5,12,6,13,10,14)(2,7,9,16,3,11,8,15)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 945);