Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| 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 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,2)(3,7)(4,8)(5,6)(10,15)(12,16)(13,22)(18,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(2,6)(3,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=fh,d^2=fh,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 | #133055, #487169, #487297, … |
| maximal normal subgroup of | #133055, #487169, #487297, … |
| maximal quotient of | #585436, #586685, #587396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51755);