Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#11999 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | #106 |
| commutator | C8 | C22×C8×C11 |
| Frattini, cosocle | C4×C8 | C23×C11 |
| socle | C22×C11 | C2×#106 |
| prime radical | C23×C11 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ C2×C4×C8×C11 ◁ G | |
234 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)(12,13,14,16,15,17,18,19)(20,21)(22,31,34,45)(23,28)(24,37)(25,40)(26,30)(27,33,42,47)(29,38)(32,44,35,49)(36,43)(39,50)(41,46,51,48);(20,22,24,33,25,35,38,48,26,34,39,47,23,32,36,46)(21,27,29,31,28,41,37,44,30,42,43,45,40,51,50,49);(20,23,26,25)(21,28,30,40)(22,32,34,35)(24,36,39,38)(27,41,42,51)(29,37,43,50)(31,44,45,49)(33,46,47,48)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 4987);