Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33×C7 · Schur multiplier (H2) = C33 |
| direct product | C33×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C33×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C33×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C7 ◁ G | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(10,11,12,13,14,15,16)(17,18,19,20)(21,24,23,22);(4,5,6)(17,21,19,23)(18,22,20,24);(7,8,9)945 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 778);