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 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,14,24)(4,8,15,22)(5,17,19,9)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27);(1,3,6,14)(2,7,10,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(9,24,17,11)(16,30,28,27)(23,32,29,25);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)Polycyclic presentation:
2^6|a^2=f,b^2=f,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 | #578⋊C3, #73139, #73140, … |
| maximal subgroup of | #2193, #2195, #3289, … |
| maximal normal subgroup of | #2193, #2195, #3289, … |
| maximal quotient of | #2898, #2913, #2947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 259);