Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C92 · Schur multiplier (H2) = C22×C9 |
| direct product | C92×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C92 | C2×D4 |
| commutator | C4 | C23×C92 |
| Frattini, cosocle | C32×C4 | C23×C32 |
| socle | C2×C32 | C2×C32×D4 |
| prime radical | C32×AGL(1,ℤ/8) | C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C92 ◁ C2×C4×C92 ◁ G | |
782 conjugacy classes of subgroups; only the 460 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 460 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 26 points:
(2,5)(3,7)(6,8)(9,10,11,12,13,14,15,16,17);(2,6)(5,8)(18,19,20,21,22,23,24,25,26);(1,2,3,5,4,6,7,8)891 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | AGL(1,ℤ/8)×C9⋊D9 |
| maximal subgroup of | AGL(1,ℤ/8)×C9⋊D9, C2×C92×AGL(1,ℤ/8) |
| maximal normal subgroup of | AGL(1,ℤ/8)×C9⋊D9, C2×C92×AGL(1,ℤ/8) |
| maximal quotient of | C92×#442, C2×C92×AGL(1,ℤ/8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5)(3,7)(6,8)(9,10,11,12,13,14,15,16,17), (2,6)(5,8)(18,19,20,21,22,23,24,25,26), (1,2,3,5,4,6,7,8));