Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = D42 (order 64) · Inn(G) = C2×D4 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72219694 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| affine general linear | AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2)(3,5)(4,6)(7,8);(2,5)(3,8)(6,7);(2,6)(5,7)Polycyclic presentation:
2^5|d^2=e,b^a=bd,c^a=ce,d^a=de,d^b=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| order | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ5 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 |
| χ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 |
| χ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 |
| χ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 |
| χ9 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Fitting subgroup of | AGL(1,ℤ/8)×A5, #14515185 |
| maximal subgroup of | #453, #454, #471, … |
| maximal normal subgroup of | #453, #454, #471, … |
| maximal quotient of | #417, #418, #428, … |
| automorphism group of | D8, Q16 |
| holomorph of | C8 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 43);