Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D8 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C22×C11 | C2×#102 |
| prime radical | C23×C11 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ C2×C4×C8×C11 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,16)(17,19)(20,21,26,31,27,32,25,42)(22,33,37,49,38,50,36,45)(23,30,24,41,40,44,39,29)(28,43,46,51,47,34,48,35);(13,16)(17,18)(20,22,24,35,26,37,40,43,27,38,39,51,25,36,23,34)(21,28,44,50,42,48,41,49,32,47,30,33,31,46,29,45);(12,14,15,19)(13,17,16,18)(20,23,25,39,27,40,26,24)(21,29,31,30,32,41,42,44)(22,34,36,51,38,43,37,35)(28,45,46,33,47,49,48,50)836 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6273);