Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | Aut(G) = C2×#483875 (order 1024) · Inn(G) = #2177 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2177 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | #508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
91 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)(3,15,18,49)(4,12)(5,23)(6,29)(7,32)(8,14)(9,17,35,56)(10,21)(11,26)(13,31)(16,51,53,64)(19,58,33,50)(20,59)(22,44)(24,43,52,48)(25,41)(27,38)(28,46)(30,47,55,39)(34,62,36,63)(37,61,45,57)(40,60)(42,54);(1,3)(2,9)(4,16)(5,24)(6,30)(7,33)(8,18)(10,34)(11,39)(12,43)(13,45)(14,35)(15,46)(17,54)(19,31)(20,57)(21,55)(22,53)(23,49)(25,61)(26,56)(27,63)(28,52)(29,36)(32,37)(38,58)(40,50)(41,51)(42,47)(44,48)(59,64)(60,62);(1,4,7,21,8,22,31,6)(2,10,13,12,14,29,32,44)(3,16,33,55,18,53,19,30)(5,25,27,26,28,20,60,54)(9,34,45,43,35,36,37,48)(11,40,41,46,42,38,59,23)(15,47,58,64,49,39,50,51)(17,24,61,63,56,52,57,62)Polycyclic presentation:
2^8|a^2=e,c^2=eg,d^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fg,g^a=gh,d^b=dfh,e^b=eh,f^b=fg,g^b=gh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151105, #151120, C2×#13657, … |
| maximal normal subgroup of | #151105, #151120, C2×#13657, … |
| maximal quotient of | #151083, #151084, #151116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6645);