Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×C8×3−1+2 |
| unambiguous semidirect product | (C8×C9)⋊(C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C7×C8 | C32 |
| commutator | C3 | C32×C7×C8 |
| Frattini, cosocle | C3×C4 | C2×C32×C7 |
| socle | C2×C3×C7 | C32×C4 |
| prime radical | C2×C32×C7 | C3×C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C7×C8 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15)(16,17,19,18,21,23,22,20,24);(17,20,21)(19,23,24)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 81);