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 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,19)(13,21)(14,15)(17,23)(18,20)(22,24)(26,31)(27,33)(32,39)(35,40);(2,5)(6,7)(9,11,14,19)(10,12,15,16)(13,18,23,22)(17,20,21,24)(25,26,30,31)(27,34,33,28)(29,32,38,39)(35,37,40,36);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(17,21)(20,24);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,16)(15,19)(17,18)(21,22)(23,24)(25,27,30,33)(26,28,31,34)(29,35,38,40)(32,36,39,37);(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,28)(26,27)(29,37)(30,34)(31,33)(32,40)(35,39)(36,38);(9,13,14,23)(10,17,15,21)(11,18,19,22)(12,20,16,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)Polycyclic presentation:
2^9|b^2=gi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154848);