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 | |
4427 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)(4,6)(5,8)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,23,24,21)(25,26,31,33)(27,32,29,39)(28,36,38,30)(34,35,37,40);(1,2,4,6)(3,5,8,7)(10,15)(11,18)(17,20)(19,24)(25,27,31,29)(26,32,33,39)(28,37,38,34)(30,35,36,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,28,31,38)(26,30,33,36)(27,34,29,37)(32,40,39,35);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,31,27)(26,32,33,39)(28,37,38,34)(30,40,36,35);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30)(26,28)(27,35)(29,40)(31,36)(32,37)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103531);