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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(11,17)(15,21)(18,23)(22,24)(26,30)(27,33)(32,36)(35,37);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,14)(15,21)(16,19)(22,24)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7890376);