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 | #2733 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 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)(3,4)(5,8)(6,7)(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,21)(22,23)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,3)(2,4)(5,6)(7,8)(9,11,15,12)(10,16,18,19)(13,22,17,20)(14,21,24,23)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(1,2,7,6)(3,5,8,4)(9,10,15,18)(11,16,12,19)(13,14,17,24)(20,23,22,21);(1,4,7,5)(2,8,6,3)(9,12,15,11)(10,16,18,19)(13,22,17,20)(14,23,24,21);(1,5,7,4)(2,8,6,3)(9,13)(10,14)(11,20)(12,22)(15,17)(16,23)(18,24)(19,21)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,6,7,2)(3,5,8,4)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,20)(19,22)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9620548);