Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 1079930880 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C210×C3 |
| abelian | C25×C3×C9 |
| multiplicative | (ℤ/3192)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C25×C32 |
| socle, prime radical | C25×C32 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
3740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2)(11,12,13,14,15,16,17,18,19);(3,4)(20,21,22);(5,6);(7,8);(9,10)864 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4050);