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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26)(27,31)(28,29)(30,34)(32,37)(33,38)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(25,27)(26,31)(28,35)(29,39)(30,37)(32,34)(33,36)(38,40);(10,15)(11,18)(17,20)(19,24)(25,27,30,37)(26,32,34,31)(28,35,38,40)(29,36,33,39);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28)(26,33)(27,35)(29,34)(30,38)(31,36)(32,39)(37,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,39)(38,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,33)(26,28,34,38)(27,36,37,39)(31,40,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650659);