Finite Group DatabasebetaFinite Group Databasebeta| order | 40 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5) (order 80) · Inn(G) = D5 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72213282 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D5 |
| commutator, hypocenter | C5 | C8 |
| Fitting, cosocle | C4×C5 | C2 |
| socle, prime radical | C2×C5 | C4 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C4 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3)(4,5)(6,7,8,10,9,11,12,13);(1,2,4,5,3)Polycyclic presentation:
2^3,5|a^2=b,b^2=c,d^a=d^416 conjugacy classes — view the full character table.
| maximal subgroup of | #674, C8×D5, #676, … |
| maximal normal subgroup of | #674, C8×D5, #676, … |
| maximal quotient of | #672, C2×#168, C3×#168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(40, 1);