Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1371 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
70 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)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21)(25,26,27);(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 507);