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 |
| prime radical | #36004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 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)(3,7)(4,5)(6,8)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(2,5)(6,7)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(26,27)(31,33)(32,34)(38,39);(1,2)(3,6)(4,5)(7,8)(11,20)(12,21)(16,23)(17,22)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625639);