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 | |
4654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,18)(11,19,21,16)(12,22,23,14)(13,17,24,20)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,18,19)(12,20,23,17)(13,22,24,14)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,8)(2,5,6,7)(9,12)(10,14)(11,20)(13,16)(15,23)(17,21)(18,22)(19,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(9,13,15,24)(10,17,18,20)(11,14,21,22)(12,16,23,19)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40);(9,14)(10,12)(11,13)(15,22)(16,20)(17,19)(18,23)(21,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=ghi,f^2=h,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, 9770308);