Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×C9×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C24 |
| commutator | C2 | C24×C3×C9 |
| Frattini, cosocle | C2×C3 | C24×C32 |
| socle | C2×C32 | C24×C3 |
| prime radical | C32×H(5,𝔽2) | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3)(13,14)(15,19)(16,17)(18,20);(4,5,6,7,8,9,10,11,12)(14,17)(18,19);(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,20)(17,19)459 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2660);