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 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,13,14)(11,17,18,15)(12,16,22,20)(19,24,21,23)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,40,37);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,18)(19,24)(21,23)(25,27,30,36)(26,31,34,35)(28,37,39,33)(29,32,40,38);(1,3,5,4)(2,7,6,8)(10,14)(11,18)(16,20)(19,21)(27,36)(28,39)(31,35)(32,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28)(26,32)(27,37)(29,35)(30,39)(31,40)(33,36)(34,38);(1,4,5,3)(2,8,6,7)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,29,30,40)(26,33,34,37)(27,38,36,32)(28,31,39,35)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,e^2=h,f^2=g,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, 7881659);