Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#33546 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C11 |
| socle | C2×C11 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×#41 ◁ G | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,17)(16,19)(20,23)(21,25)(22,24)(26,27);(13,17)(18,23)(19,24)(25,27);(15,22)(20,26);(12,14,16,21)(13,18,19,25)(15,20,22,26)(17,23,24,27)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 26534);