Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | C22 |
| commutator | C2 | C2×C3×C8×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C32 |
| socle | C22×C32 | C2×C3×C4 |
| prime radical | C23×C32 | C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
250 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(13,14)(15,16)(17,18,19,21,20,22,23,24);(4,5,6,7,8,9,10,11,12)(14,16)540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 563);