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 | |
4191 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)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(27,33)(30,34)(35,39)(37,40);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,28)(27,30,33,34)(29,36,38,32)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(12,21)(22,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|b^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8961625);