Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/8) (order 768) · Inn(G) = C22×D4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
134 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)Polycyclic presentation:
2^6|e^2=f,b^a=be,e^a=ef,e^b=ef,d^c=df22 conjugacy classes — view the full character table.
| Fitting subgroup of | #576⋊C3, #73137, #73138, … |
| maximal subgroup of | #2192, #2194, #3288, … |
| maximal normal subgroup of | #2192, #2194, #3288, … |
| maximal quotient of | #2897, #2913, #2946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 257);