Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C33 |
| direct product | C33×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | D8 |
| commutator, Frattini, cosocle | C8 | C22×C33 |
| prime radical | C33×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C33×C4 ◁ C33×C8 ◁ G | |
364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(4,5,6)(11,17)(12,24)(13,15)(14,22)(16,20)(19,25)(21,23);(7,8,9)297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2192);