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) = #72214866 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4 |
| commutator | C4 | C2×C4 |
| Frattini, cosocle | C2×C4 | C22 |
| socle | C2 | C4⋊C4 |
| prime radical | C22 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)Polycyclic presentation:
2^5|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bc,c^a=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | #108×A5, #14514975 |
| maximal subgroup of | #359, #361, #362, … |
| maximal normal subgroup of | #359, #361, #362, … |
| maximal quotient of | #334, #335, #341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 15);