Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C23 |
| commutator | C22 | C23×C3×C9 |
| Frattini, cosocle | C22×C3 | C23×C32 |
| socle | C22×C32 | C23×C3 |
| prime radical | C23×C32 | C22×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(14,18)(15,22)(20,27)(21,25)(23,26)(24,28);(4,5,6,7,8,9,10,11,12)(13,14,19,25)(15,20,26,27)(16,18,17,21)(22,28,23,24);(13,15,16,23)(14,20,18,24)(17,22,19,26)(21,28,25,27)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1880);