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 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,31)(29,37)(30,32)(33,38)(35,39)(36,40);(1,2,6,5)(3,8,7,4)(10,14)(15,18)(16,21)(22,24)(25,27,30,28)(26,31,32,34)(29,35,38,36)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(11,17)(12,20)(15,18)(16,21)(26,32)(29,38)(31,34)(35,36);(1,2,6,5)(3,4,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,39)(36,40)(37,38);(1,4,6,8)(2,7,5,3)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,27)(26,31,32,34)(29,36,38,35)(33,39,37,40);(1,5,6,2)(3,8,7,4)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,33,32,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^9|b^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837481);