Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #12687 (order 256) · Inn(G) = C23 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C22 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(3,7)(5,8)(9,10,12,14)(11,13,15,16);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^5|a^2=d,b^2=e,c^2=d,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| derived subgroup of | #7523, #7524, #7525, … |
| Frattini subgroup of | #13014548, #13014549, #13014550, … |
| Fitting subgroup of | #125×A5, #14515211, #14515267 |
| maximal subgroup of | #330, #478, #479, … |
| maximal normal subgroup of | #330, #478, #479, … |
| maximal quotient of | #382, #387, #389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 32);