Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = #72285688 (order 1024) · Inn(G) = #117 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | #2163 |
| prime radical | C23 | M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11,5,12,21,37,6,13,22,36,23,38,48,59,7,10,20,33,24,34,47,57,25,9,29,31,49,32,54,56)(3,14,16,35,17,39,42,58,18,30,43,52,44,55,61,64,15,8,19,27,40,28,45,53,41,26,46,50,60,51,62,63);(1,3)(2,8)(4,19)(5,17)(6,18)(7,15)(9,30)(10,14)(11,35)(12,28)(13,26)(16,20)(21,45)(22,46)(23,44)(24,40)(25,41)(27,33)(29,43)(31,50)(32,55)(34,39)(36,52)(37,58)(38,51)(42,47)(48,62)(49,60)(53,57)(54,61)(56,63)(59,64);(2,9,10,13)(3,15)(4,20)(8,26,14,30)(11,36,33,31)(12,32,34,38)(17,40)(18,41)(21,47)(22,29)(27,52,35,50)(28,51,39,55)(37,59,57,56)(44,60)(48,54)(53,64,58,63)Polycyclic presentation:
2^8|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,c^b=ch,e^b=eh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94003, #94004, #149731, … |
| maximal normal subgroup of | #94003, #94004, #149731, … |
| maximal quotient of | #151882, #151894, #151982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6631);