Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C27 · Schur multiplier (H2) = C23 |
| direct product | C4×D4×C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C27 | C22 |
| commutator | C2 | C22×C4×C27 |
| Frattini, cosocle | C22×C9 | C23×C3 |
| socle | C22×C3 | C23×C9 |
| prime radical | C2×C3×D4 | C2×C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C27 ◁ G | |
188 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(5,6,7,8);(6,8)540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 174);