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 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,30,31)(27,29,34,32)(28,36,37,33)(35,40,38,39);(2,5)(3,8)(10,14)(15,17)(16,20)(23,24)(26,31)(28,37)(29,32)(35,38);(1,3)(2,6)(4,5)(7,8)(11,18)(15,17)(19,21)(23,24)(26,31)(27,34)(33,36)(35,38);(1,4,7,6)(2,3,5,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,27)(26,32)(28,38)(29,31)(30,34)(33,39)(35,37)(36,40);(1,5,7,2)(3,4,8,6)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,28,30,37)(26,33,31,36)(27,35,34,38)(29,39,32,40);(1,6,7,4)(2,8,5,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29,30,32)(26,27,31,34)(28,39,37,40)(33,35,36,38)Polycyclic presentation:
2^9|a^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8745639);