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×#509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C19 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C8×C19 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,25)(23,32)(24,26)(27,29)(28,30)(31,33)(34,35);(21,25)(22,29)(23,30)(26,33)(27,28)(31,35)(32,34);(21,26)(25,33)(27,34)(28,32)304 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 190);