Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #511790 (order 512) · Inn(G) = C23 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C22 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,9,12)(7,11,8,10);(5,7,9,8)(6,10,12,11);(1,2,3,4)(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^5|a^2=d,b^2=d,c^2=e,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| derived subgroup of | #7519, #7520, #7521, … |
| Frattini subgroup of | #1405, #1410, #1411, … |
| Fitting subgroup of | C4⋊Q8×A5, C4⋊#91883, #14515210 |
| maximal subgroup of | #332, #333, #356, … |
| maximal normal subgroup of | #332, #333, #356, … |
| maximal quotient of | #384, #389, #391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 35);