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 | #35538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,20)(13,15)(14,19)(17,22)(18,23)(21,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(3,7)(5,8)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(14,16)(17,20)(23,24)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9088144);