Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = AGL(2,ℤ/4) (order 1536) · Inn(G) = C23 · Out(G) = C2×#1020 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 8 points:
(2,4)(6,7);(5,6,8,7);(1,2,3,4)(5,7,8,6)Polycyclic presentation:
2^5|b^2=d,c^2=e,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| derived subgroup of | #7515, #7516, #7517, … |
| Frattini subgroup of | #1402, #1408, #1409, … |
| Fitting subgroup of | #127⋊C3, #4626, #857⋊C2, … |
| maximal subgroup of | #331, #353, #453, … |
| maximal normal subgroup of | #331, #353, #453, … |
| maximal quotient of | #384, #390, #392, … |
| outer automorphism group of | #11538 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 34);