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 | |
682 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)(3,7,8,5)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,32)(28,33)(29,36)(30,31)(34,39)(35,38)(37,40);(1,2,4,6)(3,5,8,7)(10,12)(15,17)(16,20)(22,23)(26,31)(27,35)(28,37)(29,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,35)(26,32,31,38)(28,34,37,36)(29,33,39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^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 | #132988, #487157, #487285, … |
| maximal normal subgroup of | #132988, #487157, #487285, … |
| maximal quotient of | #585270, #585922, #587536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46963);