Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C2 | C24×C3×C4 |
| Frattini, cosocle | C22 | C25×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(3,4,5);(6,7,8,9)(11,15)(13,21)(16,18)(19,25);(10,11,14,15)(12,18,20,16)(13,17,21,22)(19,24,25,23);(10,12,14,20)(11,16,15,18)(13,19,21,25)(17,23,22,24);(1,2)(10,13)(11,17)(12,19)(14,21)(15,22)(16,23)(18,24)(20,25)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19886);