Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(1,2)(3,7)(4,5)(6,8)(10,15)(12,16)(18,21)(20,23)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(2,5)(6,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53768, #8103364, #8103473, … |
| maximal normal subgroup of | C2×#53768, #8103364, #8103473, … |
| maximal quotient of | #586026, #586693, #587172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46756);