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 | |
4459 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(2,6)(3,7)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(26,32)(27,28)(34,37)(35,36);(1,3,5,7)(2,4,6,8)(10,17)(14,21)(16,18)(20,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,13,15,11)(10,18,17,16)(12,22,23,19)(14,24,21,20)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=ghi,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, 8104325);