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 | #47991 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(5,7)(25,26,31,35)(27,28,36,32)(29,33,38,40)(30,34,37,39);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,36)(26,32,35,28)(29,37,38,30)(33,34,40,39);(1,3,4,8)(2,5,6,7)(10,14)(12,21)(15,17)(18,22)(29,38)(30,37)(33,40)(34,39);(2,6)(5,7)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,31,32)(26,27,35,36)(29,39,38,34)(30,33,37,40);(1,2)(3,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,33)(27,30)(28,34)(31,38)(32,39)(35,40)(36,37);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30)(26,34)(27,29)(28,33)(31,37)(32,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947438);