Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #23900 (order 256) · Inn(G) = C23 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C23 |
| prime radical | C2×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(7,11)(9,12);(1,2,4,3)(5,6,8,10)(7,9,11,12);(1,3,4,2)(5,7)(6,9)(8,11)(10,12)Polycyclic presentation:
2^5|b^2=e,c^2=d,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| derived subgroup of | #13677, #13678, #13679, … |
| Frattini subgroup of | #1403, #1404, #1406, … |
| Fitting subgroup of | #124×A5, #14515179, #14515268 |
| maximal subgroup of | #329, #354, #355, … |
| maximal normal subgroup of | #329, #354, #355, … |
| maximal quotient of | #382, #388, #390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 31);