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×C9×2−1+4 |
| 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 | C32×2−1+4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3)(14,18)(16,24)(19,21)(22,28);(4,5,6,7,8,9,10,11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)459 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2663);