Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D4 |
| commutator | C4 | C42×C5 |
| Frattini, cosocle | C22×C4 | C22×C5 |
| socle | C22×C5 | C4⋊C4 |
| prime radical | C23×C5 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 25 points:
(5,6,8,13,10,14,9,12)(7,15,16,11,18,19,17,20)(21,22,23,24,25);(1,2,3,4)(5,7,9,17,10,18,8,16)(6,11,13,19,14,20,12,15)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 146);