Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1368 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #106 |
| commutator | C8 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(10,14,15,20)(11,17)(12,19)(16,23,22,21)(18,24)(25,26,27);(9,10,11,14,12,16,17,21,13,15,18,20,19,22,24,23)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 504);