Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#578 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C19 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ #41×C19 ◁ G | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,29)(22,30,33,43)(23,27,34,41)(24,36,38,28)(26,37,40,32)(31,44,45,51)(35,48,47,42)(39,49,50,46);(20,22,25,33)(21,26,29,40)(23,31,34,45)(24,37,38,32)(27,39,41,50)(28,43,36,30)(35,49,47,46)(42,51,48,44);(20,23)(21,27)(22,31)(24,35)(25,34)(26,39)(28,42)(29,41)(30,44)(32,46)(33,45)(36,48)(37,49)(38,47)(40,50)(43,51);(23,34)(27,41)(31,45)(35,47)(39,50)(42,48)(44,51)(46,49)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 259);