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 | #38836 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4303 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)(3,7)(4,5)(6,8)(10,15)(11,18)(17,24)(19,20)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(2,5)(6,7)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,2)(3,6)(4,5)(7,8)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=i,e^2=ghi,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, 10124036);