Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C22⋊C4 |
| commutator | C22 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C32 |
| socle | C2×C32 | C3×C22⋊C4 |
| prime radical | C22×C32 | C2×C3×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C3×C4×C9 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3)(13,14,16,20,17,19,25,27)(15,21,22,28,23,18,24,26);(4,5,6,7,8,9,10,11,12)(13,15,17,23)(14,18,19,21)(16,24,25,22)(20,28,27,26)297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 572);