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 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,14)(15,21)(16,19)(22,24)(25,26,31,34)(27,35,37,30)(28,33,29,32)(36,39,38,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,30)(28,36)(29,38)(31,37)(32,39)(33,40)(34,35);(1,3)(2,4)(5,8)(6,7)(11,17)(15,21)(18,23)(22,24)(25,28,31,29)(26,32,34,33)(27,36,37,38)(30,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,29,31,28)(26,33,34,32)(27,36,37,38)(30,39,35,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27)(26,30)(28,38)(29,36)(31,37)(32,40)(33,39)(34,35);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,27)(28,39)(29,40)(31,35)(32,36)(33,38)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8429333);