Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C19 · Schur multiplier (H2) = C29 |
| direct product | C19×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C19 |
| socle | C2×C19 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 5 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)(22,28)(23,27)(24,26)(25,29)(30,34)(31,33)(32,35);(20,22)(21,25)(23,30)(24,29)(26,28)(27,32)(31,35)(33,34);(22,29)(25,28)(30,35)(32,34);(21,26)(25,28)(27,33)(32,34);(20,23,24,31)(21,27,26,33)(22,30,29,35)(25,32,28,34)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 266);