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 | #39388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,38,40,39);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,4,6,3)(2,7,5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613537);