Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#367 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | D8 |
| commutator | C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4)(6,10)(7,14)(8,15)(11,16)(12,20)(13,19)(17,18)(21,22,23,24,25);(5,6,11,20,8,12,7,10,9,13,14,18,15,17,16,19)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 172);