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 | C2×C19×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C19 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 5 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);(3,4)(5,9)(6,7)(8,10);(4,7)(8,9);(3,4)(5,8)(6,7)(9,10);(3,5)(4,8)(6,10)(7,9)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 264);