Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C24 |
| commutator | C2 | C23×C4×C5 |
| Frattini, cosocle | C4 | C24×C5 |
| socle | C2×C5 | C23×C4 |
| prime radical | C2×C5×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,19,20,21);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16);(2,7)(6,12)(8,14)(13,16)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1570);