Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C19 |
| prime radical | C19×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C8×C19 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,30)(23,34)(24,39)(25,27)(26,31)(28,37)(29,36)(32,50)(33,47)(35,45)(38,46)(40,44)(41,49)(42,51)(43,48);(20,22,25,33)(21,26,27,41)(23,35,38,51)(24,40,36,32)(28,44,43,50)(29,45,39,42)(30,46,47,34)(31,48,49,37);(21,27)(26,41)(28,43)(29,39)(30,47)(34,46)(42,45)(44,50)304 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 191);