Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C5 |
| prime radical | C2×C5×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4,5,6,7);(9,13)(10,16)(11,18)(14,21)(15,19)(20,22);(8,9,12,15)(10,14,16,22)(11,19,18,13)(17,21,23,20);(1,2)(8,10)(9,14)(11,17)(12,16)(13,20)(15,22)(18,23)(19,21)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1576);