Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#120 (order 128) · Inn(G) = C23 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
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 | |
47 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)(6,8);(5,6,7,8);(1,2)(3,4)Polycyclic presentation:
2^5|b^2=d,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | #121×A5, #14515175, #14515176, … |
| maximal subgroup of | #327, #447, #448, … |
| maximal normal subgroup of | #327, #447, #448, … |
| maximal quotient of | #380, #386, #388, … |
| outer automorphism group of | #7360, #7361, #7369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 28);