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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,22)(21,23)(25,26,31,32)(27,29,33,37)(28,30,36,39)(34,35,38,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27,31,33)(26,29,32,37)(28,34,36,38)(30,35,39,40);(1,3,5,6)(2,4,7,8)(10,12)(16,18)(17,20)(23,24)(26,32)(27,33)(28,36)(35,40);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28)(26,30)(27,34)(29,35)(31,36)(32,39)(33,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,37)(26,33,32,27)(28,35,36,40)(30,38,39,34);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,30)(26,28)(27,35)(29,34)(31,39)(32,36)(33,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9589759);