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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(17,24)(19,20)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(2,5)(6,7)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,2)(3,6)(4,5)(7,8)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123944);