Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator | C2 | C23×C4×C19 |
| Frattini, cosocle | C22 | C24×C19 |
| prime radical | C23×C19 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(25,29)(27,35)(30,32)(33,39);(24,25,28,29)(26,32,34,30)(27,31,35,36)(33,38,39,37);(24,26,28,34)(25,30,29,32)(27,33,35,39)(31,37,36,38);(24,27)(25,31)(26,33)(28,35)(29,36)(30,37)(32,38)(34,39)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 200);