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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(9,10)(11,16)(12,13)(14,23)(15,19)(17,22)(18,20)(21,24)(26,31)(27,34)(29,37)(35,39);(1,3)(2,4)(5,6)(7,8)(9,11,15,21)(10,16,19,24)(12,20,22,14)(13,18,17,23)(25,26,30,31)(27,32,34,38)(28,29,33,37)(35,36,39,40);(9,12,15,22)(10,17,19,13)(11,14,21,20)(16,18,24,23)(25,27,30,34)(26,32,31,38)(28,35,33,39)(29,36,37,40);(1,4,7,5)(2,8,6,3)(9,12)(10,13)(11,20)(14,21)(15,22)(16,18)(17,19)(23,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,5,7,4)(2,8,6,3)(9,13)(10,12)(11,18)(14,24)(15,17)(16,20)(19,22)(21,23)(25,28)(26,29)(27,35)(30,33)(31,37)(32,36)(34,39)(38,40);(1,6,7,2)(3,5,8,4)(9,14,15,20)(10,18,19,23)(11,12,21,22)(13,16,17,24)(25,29)(26,33)(27,36)(28,31)(30,37)(32,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770640);