Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C19 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(22,28)(23,31)(25,29)(26,33)(36,37,40,43)(38,41,46,44)(39,47,48,42)(45,50,51,49);(20,21,24,27)(22,29,28,25)(23,26,31,33)(30,35,34,32)(36,37)(38,44)(39,47)(40,43)(41,46)(42,48)(45,49)(50,51);(20,22,24,28)(21,25,27,29)(23,30,31,34)(26,32,33,35)(36,38)(37,41)(39,45)(40,46)(42,49)(43,44)(47,50)(48,51);(20,23)(21,26)(22,30)(24,31)(25,32)(27,33)(28,34)(29,35)(36,39,40,48)(37,42,43,47)(38,45,46,51)(41,49,44,50)361 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 245);