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 | |
4561 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)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,40,36,34);(2,5)(6,7)(10,14)(11,17)(16,21)(19,23)(25,27,31,38)(26,32,30,35)(28,36,37,29)(33,40,39,34);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(19,23)(22,24)(26,30)(32,35)(33,39)(34,40);(1,3,4,8)(2,6,5,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,33)(27,36)(29,38)(30,39)(31,37)(32,40)(34,35);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,39,35,33);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,40,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8325370);