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 | #35156 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(11,16)(17,21)(18,24)(25,26,31,33)(27,37,38,32)(28,30,36,34)(29,39,40,35);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,12,16,19)(13,21,23,17)(18,22,24,20)(25,27)(26,32)(28,35)(29,34)(30,40)(31,38)(33,37)(36,39);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(17,21)(20,22)(26,33)(27,38)(28,36)(29,40);(1,4,5,3)(2,8,6,7)(9,11,14,16)(10,12,15,19)(13,18,23,24)(17,20,21,22)(25,28,31,36)(26,34,33,30)(27,35,38,39)(29,37,40,32);(9,12)(10,16)(11,15)(13,22)(14,19)(17,18)(20,23)(21,24)(25,29,31,40)(26,35,33,39)(27,34,38,30)(28,37,36,32);(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,20,19,22)(25,30,31,34)(26,36,33,28)(27,29,38,40)(32,35,37,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893421);