Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#41 (order 32) · Inn(G) = C22 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72213945 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C8 | C22 |
| commutator, socle | C2 | C2×C8 |
| prime radical | C22 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
13 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(2,10)(4,12)(6,14)(8,16)Polycyclic presentation:
2^5|a^2=c,c^2=d,d^2=e,b^a=be20 conjugacy classes — view the full character table.
| Fitting subgroup of | M32×A5, #14514993 |
| maximal subgroup of | #347, #349, #350, … |
| maximal normal subgroup of | #347, #349, #350, … |
| maximal quotient of | #346, C22⋊C16, C4⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 17);