Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
90 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)(3,6)(4,13)(5,7)(8,10)(9,11)(12,14)(15,16)(17,18,19,20,21);(2,6)(3,10)(4,11)(7,14)(8,9)(12,16)(13,15);(2,7)(6,14)(8,15)(9,13)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1010);