Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C22×#442 (order 256) · Inn(G) = D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D8 |
| commutator | C8 | C2×C4 |
| Frattini, cosocle | C2×C8 | C22 |
| prime radical | C2×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,17)(9,13)(11,14)(12,15)(16,19)(18,20);(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18)Polycyclic presentation:
2^6|a^2=d,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,c^b=ce,e^b=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#357, #2138, #2139, … |
| maximal normal subgroup of | C2×#357, #2138, #2139, … |
| maximal quotient of | #1331, #1347, #1361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 38);