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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(13,21)(17,22)(18,23)(20,24)(25,26,32,36)(27,28,30,33)(29,34,37,40)(31,38,39,35);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27,32,30)(26,33,36,28)(29,31,37,39)(34,35,40,38);(1,3,4,7)(2,6,5,8)(10,15)(12,19)(17,22)(20,24)(25,28,32,33)(26,27,36,30)(29,35,37,38)(31,34,39,40);(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,29)(26,34)(27,31)(28,38)(30,39)(32,37)(33,35)(36,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,30,32,27)(26,28,36,33)(29,39,37,31)(34,38,40,35);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,31,32,39)(26,35,36,38)(27,37,30,29)(28,34,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619844);