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 | #2676 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(25,26)(27,36)(28,38)(29,37)(30,33)(31,32)(34,39)(35,40);(2,6)(3,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,21)(23,24)(26,32)(28,39)(33,36)(35,37);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(18,24)(19,21)(25,27,31,30)(26,33,32,36)(28,35,39,37)(29,34,40,38);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,28,31,39)(26,34,32,38)(27,37,30,35)(29,36,40,33);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,29)(26,35)(27,34)(28,36)(30,38)(31,40)(32,37)(33,39);(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,30,31,27)(26,33,32,36)(28,37,39,35)(29,34,40,38)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633873);