Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×#1370 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | D4 |
| commutator | C4 | C2×C3×C16 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,5)(3,6)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24)(25,26,27);(1,2,6,8,4,7,3,5)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 506);