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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
716 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(2,3)(4,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,2)(4,7,8,6)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fg,d^2=g,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×#54301, #8103709, #8103891, … |
| maximal normal subgroup of | C2×#54301, #8103709, #8103891, … |
| maximal quotient of | #585356, #585914, #586938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47289);