Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #39118 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,19,21,15)(12,16,22,23)(13,17,20,24)(25,26,31,36)(27,32,29,34)(28,38,39,33)(30,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,19)(12,20,22,13)(16,17,23,24)(25,27,31,29)(26,32,36,34)(28,30,39,37)(33,40,38,35);(2,5)(3,7)(12,22)(13,20)(16,23)(17,24)(25,28,31,39)(26,33,36,38)(27,37,29,30)(32,35,34,40);(3,7)(6,8)(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,21)(20,24)(25,29,31,27)(26,34,36,32)(28,37,39,30)(33,35,38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,21)(14,22)(15,17)(18,23)(19,24)(25,30,31,37)(26,35,36,40)(27,28,29,39)(32,38,34,33);(1,2)(3,6)(4,5)(7,8)(9,13,14,20)(10,17,18,24)(11,12,21,22)(15,23,19,16)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9422553);