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 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,16)(11,14)(12,22)(21,24)(29,37)(32,36)(34,40)(38,39);(2,5)(3,7)(9,10)(11,19)(12,13)(14,17)(15,16)(18,22)(20,21)(23,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,17,16,19)(12,20,22,23)(13,24,18,21)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,20)(13,16)(14,23)(15,22)(17,21)(19,24);(9,13,15,18)(10,12,16,22)(11,21,14,24)(17,23,19,20)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,14,15,11)(10,17,16,19)(12,23,22,20)(13,24,18,21)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002566);