Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C25×C4 |
| direct product | C11×#10935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | #120 |
| commutator | C23 | C2×C42×C11 |
| Frattini, cosocle | C25 | C23×C11 |
| prime radical | C26×C11 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C25×C11 ◁ G | |
2594 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 2594 conjugacy classes of subgroups — too many to draw. Only its 302 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) = 352 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(20,21)(22,26,27,24)(23,25);(16,17,18,19)(20,22)(21,24)(23,27)(25,26);(12,13,14,15)(20,21)(22,24)(23,25)(26,27)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 3923);