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 | |
4312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,18)(19,23)(22,24)(25,26,30,31)(27,34,37,32)(28,29,38,33)(35,39,40,36);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(26,31)(29,33)(32,34)(36,39);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(22,24)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,39)(38,40);(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28)(26,33)(27,35)(29,31)(30,38)(32,39)(34,36)(37,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663331);