Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 23328 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C9 |
| abelian | C22×C92 |
| multiplicative | (ℤ/513)ˣ, (ℤ/1026)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C32 | C22×C32 |
| socle, prime radical | C22×C32 | C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(5,6,7,8,9,10,11,12,13);(3,4)(14,15,16,17,18,19,20,21,22)324 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 81);