Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C19×#127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ G | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29);(4,6)(8,9);(7,8,10,9);(3,4,5,6)(7,9,10,8)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 211);