Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D4 |
| commutator | C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| socle | C22×C5 | C2×D4 |
| prime radical | C23×C5 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22,23,24,25);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 928);