Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = AGL(1,ℤ/32) (order 512) · Inn(G) = D16 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D16 |
| commutator, Frattini, cosocle | C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
17 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)Polycyclic presentation:
2^6|c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,c^b=cd,d^b=de,e^b=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | D64, SD128, C2×D32, … |
| maximal normal subgroup of | D64, SD128, C2×D32, … |
| maximal quotient of | #1415, #1423, D64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 52);