Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C7 · Schur multiplier (H2) = C22 |
| direct product | C5×C7×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C5×C7 |
| socle | C2×C5×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C7 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14)(15,18)(16,17)(19,20);(14,17)(18,20);(13,15,16,19)(14,18,17,20)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 169);