Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #511819 (order 512) · Inn(G) = C2×D4 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24 |
| socle | C2 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(2,8)(7,15)(9,13)(10,16);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)Polycyclic presentation:
2^6|d^2=f,e^2=f,b^a=be,c^a=cf,e^a=ef,e^b=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1881, #1906, #2008, … |
| maximal normal subgroup of | #1881, #1906, #2008, … |
| maximal quotient of | #2896, #2911, #2943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 256);