Finite Group DatabasebetaFinite Group Databasebeta| order | 8 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22 (order 4) · Inn(G) = C1 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = AGL(1,ℤ/8) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C2 |
| socle, prime radical | C2 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 1 generator on 8 points:
(1,2,3,4,5,6,7,8)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| order | 1 | 8 | 4 | 8 | 2 | 8 | 4 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | ζ | ζ3 | -1 | −ζ | −ζ3 | ||
| χ3 | 1 | -1 | 1 | -1 | ||||
| χ4 | 1 | ζ3 | ζ | -1 | −ζ3 | −ζ | ||
| χ5 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
| χ6 | 1 | −ζ | −ζ3 | -1 | ζ | ζ3 | ||
| χ7 | 1 | -1 | 1 | -1 | ||||
| χ8 | 1 | −ζ3 | −ζ | -1 | ζ3 | ζ |
Here ζ is a primitive root of unity of order 8 (ζ8 = 1); i = ζ2.
| center of | M32, #131, C3⋊C16, … |
| derived subgroup of | D16, SD32, Q32, … |
| Frattini subgroup of | C16, C2×C16, M32, … |
| Fitting subgroup of | C8×A5, #91145, A5⋊C16, … |
| maximal subgroup of | C16, C2×C8, M16, … |
| maximal normal subgroup of | C16, C2×C8, M16, … |
| maximal quotient of | C16, C2×C8, C3⋊C8, … |
| outer automorphism group of | D17, SD64, #3928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8, 1);