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 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,38)(29,31)(30,34)(33,39)(35,40)(36,37);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,30)(29,36)(31,37)(33,40)(34,38)(35,39);(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,28,31,39)(26,33,29,38)(27,35,37,30)(32,34,36,40);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,28)(26,33)(27,30)(29,38)(31,39)(32,40)(34,36)(35,37);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,31,26)(27,36,37,32)(28,38,39,33)(30,34,35,40);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30)(26,34)(27,28)(29,40)(31,35)(32,38)(33,36)(37,39)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809527);