Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C2×C5×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(3,4,5,6,7);(9,13)(11,19)(14,16)(17,23);(8,9,12,13)(10,16,18,14)(11,15,19,20)(17,22,23,21);(8,10,12,18)(9,14,13,16)(11,17,19,23)(15,21,20,22);(1,2)(8,11)(9,15)(10,17)(12,19)(13,20)(14,21)(16,22)(18,23)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1633);