Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×D4 |
| commutator | C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| socle | C2×C5 | C2×C22⋊C4 |
| prime radical | C2×C5×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,5,8,6,9,4,13)(3,10,12,7,11,16,14,15)(17,18,19,20,21);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13);(3,11)(7,15)(10,16)(12,14)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 922);