Finite Group DatabasebetaFinite Group Databasebeta| order | 1232 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C11 · Schur multiplier (H2) = C2 |
| direct product | C7×C11×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7×C11 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C7×C11 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C11 ◁ C4×C7×C11 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 616 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26);(20,26)(21,25)(22,24)539 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1232, 83);