Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/8) (order 64) · Inn(G) = D4 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72219681 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4 |
| commutator, Frattini, cosocle | C4 | C23 |
| socle | C2 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)Polycyclic presentation:
2^5|c^2=e,d^2=e,b^a=bd,d^a=de,d^b=de14 conjugacy classes — view the full character table.
| Fitting subgroup of | #135×A5, #14515183, #72277090 |
| maximal subgroup of | #359, #360, #443, … |
| maximal normal subgroup of | #359, #360, #443, … |
| maximal quotient of | #416, #427, C4×D8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 42);