Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C13 · Schur multiplier (H2) = C22 |
| direct product | C3×C13×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C13 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C3×C13 |
| socle | C2×C3×C13 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C13 ◁ C2×C3×C4×C13 ◁ G | |
124 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)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)546 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1149);