Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C13 · Schur multiplier (H2) = C1 |
| direct product | C3×C13×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C13 | D4 |
| commutator | C4 | C2×C3×C4×C13 |
| Frattini, cosocle | C2×C4 | C22×C3×C13 |
| socle | C2×C3×C13 | C4⋊C4 |
| prime radical | C22×C3×C13 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C13 ◁ C2×C3×C4×C13 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 624 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,21,24,22,25,20,30)(19,26,28,31,29,32,27,23);(17,19,20,27,22,29,21,28)(18,23,24,26,25,31,30,32)546 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 275);