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 | |
4472 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)(9,10)(11,16)(12,21)(13,14)(15,17)(18,24)(19,23)(20,22)(26,31)(27,35)(33,37)(34,36);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,16,17,21)(13,19,24,22)(14,23,18,20)(25,26,30,31)(27,32,35,39)(28,33,38,37)(29,36,40,34);(1,3,5,4)(2,7,6,8)(10,17)(11,12)(13,24)(20,23)(26,31)(28,38)(32,39)(34,36);(9,12,15,11)(10,16,17,21)(13,22,24,19)(14,23,18,20)(25,27)(26,32)(28,36)(29,37)(30,35)(31,39)(33,40)(34,38);(9,13,15,24)(10,18,17,14)(11,19,12,22)(16,20,21,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,40)(34,35);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,13,17,24)(11,20,12,23)(16,22,21,19)(25,29,30,40)(26,34,31,36)(27,37,35,33)(28,32,38,39)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=gh,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, 7881027);