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 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,6,4)(2,7,5,8)(11,20)(12,22)(15,19)(16,23)(26,30)(31,33)(32,36)(38,39);(9,11,14,20)(10,15,18,19)(12,21,22,13)(16,17,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(2,5)(7,8)(9,12)(10,16)(11,21)(13,20)(14,22)(15,17)(18,23)(19,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,4,6,3)(2,8,5,7)(9,13,14,21)(10,17,18,24)(11,12,20,22)(15,23,19,16);(1,5,6,2)(3,8,4,7)(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343983);