Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×D4 (order 32) · Inn(G) = D4 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72214082 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4 |
| commutator | C4 | C2×C4 |
| Frattini, cosocle | C2×C4 | C22 |
| socle | C2 | C22⋊C4 |
| prime radical | #41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2,4,7,5,3,8,6);(1,3)(2,5)(4,6)(7,8)Polycyclic presentation:
2^5|a^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,c^b=ce14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 11);