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 | |
4505 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)(11,20)(12,21)(13,14)(15,17)(16,22)(18,24)(19,23)(25,26)(27,30)(28,31)(29,33)(32,35)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,20)(12,23,24,14)(13,18,19,21)(25,27)(26,30)(28,36)(29,35)(31,40)(32,33)(34,38)(37,39);(2,5)(6,8)(10,17)(13,19)(16,20)(18,21)(25,26,29,33)(27,32,35,30)(28,31,38,37)(34,40,36,39);(2,5)(3,7)(9,12,15,24)(10,18,17,21)(11,14,22,23)(13,20,19,16)(25,28)(26,31)(27,34)(29,38)(30,39)(32,40)(33,37)(35,36);(1,2)(3,6)(4,5)(7,8)(9,13,15,19)(10,14,17,23)(11,21,22,18)(12,16,24,20)(25,26)(27,32)(28,37)(29,33)(30,35)(31,38)(34,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,14)(10,19)(11,12)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,32)(28,36)(29,35)(30,33)(31,39)(34,38)(37,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8986000);