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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,5)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,2)(3,7)(4,6)(5,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,4,6,2)(3,7,5,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(1,5,6,3)(2,7,4,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fh,d^2=f,e^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×#56871, #8099948, #8100312, … |
| maximal normal subgroup of | C2×#56871, #8099948, #8100312, … |
| maximal quotient of | #585595, #586412, #586951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49859);