Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C11 · Schur multiplier (H2) = C213 |
| direct product | C22×C11×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C11 | C23 |
| commutator | C22 | C24×C4×C11 |
| Frattini, cosocle | C23 | C25×C11 |
| prime radical | C27×C11 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C11 ◁ G | |
33214 conjugacy classes of subgroups; only the 3614 normal subgroups are shown.
This group has 33214 conjugacy classes of subgroups — too many to draw. Only its 3614 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(17,18,19,20,21,22,23,24,25,26,27);(3,4);(6,8)(9,10,11,12)(14,16);(13,14)(15,16);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)1232 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 26991);