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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,17)(11,21,22,16)(12,18,19,13)(14,20,23,24)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,17,21)(12,14,19,23)(13,20,18,24)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,3,4,7)(2,5,6,8)(10,17)(11,22)(12,19)(20,24)(26,33)(28,34)(32,36)(35,40);(9,12,15,19)(10,18,17,13)(11,14,22,23)(16,24,21,20)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(9,14)(10,20)(11,19)(12,22)(13,21)(15,23)(16,18)(17,24)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7796124);