Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×#41 (order 192) · Inn(G) = C1 · Out(G) = C22×C3×#41 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C9×C16 |
| multiplicative | (ℤ/323)ˣ, (ℤ/459)ˣ, (ℤ/646)ˣ, (ℤ/918)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C3×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(19,20,21,22,23,24,25,26,27);(17,18)288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 59);