Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C13 · Schur multiplier (H2) = C2 |
| direct product | C3×C13×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C13 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C3×C13 |
| prime radical | C23×C3×C13 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C13 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 624 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16)(18,22)(19,26)(24,31)(25,29)(27,30)(28,32);(17,18,23,29)(19,24,30,31)(20,22,21,25)(26,32,27,28);(17,19,20,27)(18,24,22,28)(21,26,23,30)(25,32,29,31)546 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1140);