Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C11×C32 · Schur multiplier (H2) = C1 |
| direct product | C11×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C32 | D4 |
| commutator | C4 | C2×C11×C32 |
| Frattini, cosocle | C2×C32 | C22×C11 |
| socle | C2×C11 | C4⋊C32 |
| prime radical | C22×C11 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C11×C32 ◁ C2×C11×C32 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,23,17,24,35,48,18,25,36,49,39,51,62,70,19,22,37,45,40,46,63,67,31,47,57,68,58,69,73,75,20,26,38,50,41,52,64,71,28,21,42,43,55,44,65,66,15,32,33,59,34,60,61,74,14,27,29,53,30,54,56,72);(12,14,15,28,20,31,19,18)(13,21,22,27,26,25,32,47)(16,29,33,42,38,57,37,36)(17,30,34,55,41,58,40,39)(23,43,45,53,50,49,59,68)(24,44,46,54,52,51,60,69)(35,56,61,65,64,73,63,62)(48,66,67,72,71,70,74,75)1232 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 532);