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 | #51423 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,26,31,33)(27,35,29,32)(28,34,38,30)(36,37,39,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,31,29)(26,32,33,35)(28,36,38,39)(30,37,34,40);(10,14)(11,17)(16,19)(18,23)(26,33)(28,38)(32,35)(36,39);(1,2,5,4)(3,6,8,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,40)(35,37);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,27)(26,35,33,32)(28,39,38,36)(30,40,34,37);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,34)(26,28,33,38)(27,37,29,40)(32,36,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7648705);