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 | #56320 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4244 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)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24)(25,26)(27,31)(28,32)(29,40)(30,34)(33,37)(35,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,14)(15,20)(17,22)(18,21)(23,24)(25,27,30,36)(26,31,34,35)(28,37,39,29)(32,33,38,40);(1,2,4,6)(3,7,8,5)(13,23)(14,24)(17,21)(18,22)(25,26,30,34)(27,35,36,31)(28,38,39,32)(29,40,37,33);(3,8)(5,7)(9,12,15,10)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,36)(37,40);(1,3,4,8)(2,5,6,7)(9,13)(10,17)(11,14)(12,21)(15,23)(16,22)(18,19)(20,24)(25,28)(26,32)(27,29)(30,39)(31,40)(33,35)(34,38)(36,37);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,13)(12,22)(15,24)(16,21)(17,19)(20,23)(25,29)(26,33)(27,28)(30,37)(31,38)(32,35)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9430958);