Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,18)(16,21)(22,24)(25,26,31,34)(27,30,29,32)(28,36,39,33)(35,40,37,38);(10,14)(11,17)(12,20)(22,24)(25,27,31,29)(26,32,34,30)(28,37,39,35)(33,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,28,31,39)(26,33,34,36)(27,35,29,37)(30,38,32,40);(1,3,4,7)(2,5,6,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,29,31,27)(26,32,34,30)(28,35,39,37)(33,38,36,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,32)(26,27,34,29)(28,38,39,40)(33,35,36,37)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50224, #7706031, #7706215, … |
| maximal normal subgroup of | C2×#50224, #7706031, #7706215, … |
| maximal quotient of | #561186, #561241, #562401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43212);