Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = #12067 (order 256) · Inn(G) = D16 · Out(G) = C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| semi dihedral | SD64 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D16 |
| commutator, Frattini, cosocle | C16 | C22 |
| prime radical | D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
16 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);(2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)Polycyclic presentation:
2^6|a^2=f,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 | C2×SD64, #2262, #2263, … |
| maximal normal subgroup of | C2×SD64, #2262, #2263, … |
| maximal quotient of | #1415, #1416, #1424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 53);