Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4×D4 (order 64) · Inn(G) = C1 · Out(G) = C2×C4×D4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#23882 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C5×C8 |
| multiplicative | (ℤ/123)ˣ, (ℤ/164)ˣ, (ℤ/246)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C22×C5 |
| socle, prime radical | C22×C5 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3,4,5,6,7,8)(11,12,13,14,15);(9,10)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 23);