Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C5 |
| prime radical | C5×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,19,20,21);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 959);