Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #61451 (order 256) · Inn(G) = C2×D8 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,13)(5,7)(8,10)(9,11)(12,14)(15,16);(2,6)(3,10)(4,11)(7,14)(8,9)(12,16)(13,15);(2,7)(6,14)(8,15)(9,13)Polycyclic presentation:
2^6|d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,d^b=de,e^b=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/16), #2190, #2192, … |
| maximal normal subgroup of | AGL(1,ℤ/16), #2190, #2192, … |
| maximal quotient of | #2139, #2140, #2159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 190);