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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26,30,32)(27,34,37,31)(28,29,39,33)(35,36,38,40);(1,2)(3,7)(4,6)(5,8)(10,11)(15,17)(16,18)(23,24)(25,27,30,37)(26,31,32,34)(28,38,39,35)(29,36,33,40);(1,3)(2,5)(4,8)(6,7)(26,32)(28,39)(31,34)(35,38);(2,6)(5,7)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,28)(26,33)(27,35)(29,32)(30,39)(31,36)(34,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,34)(28,38)(29,40)(30,37)(31,32)(33,36)(35,39);(1,3)(2,7)(4,8)(5,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,33)(26,28,32,39)(27,36,37,40)(31,38,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=gh,f^2=g,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, 10126657);