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 | #40514 | 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:
(1,2,5,3)(4,7,8,6)(25,26)(27,32)(28,30)(29,34)(31,33)(35,36)(37,40)(38,39);(2,3)(6,7)(9,10,14,15)(11,17,20,16)(12,21,23,13)(18,24,22,19)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,3,5,2)(4,7,8,6)(10,15)(13,21)(16,17)(19,24)(26,33)(27,34)(30,40)(35,39);(1,4)(2,6)(3,7)(5,8)(9,11,14,20)(10,16,15,17)(12,22,23,18)(13,19,21,24)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(9,12)(10,13)(11,18)(14,23)(15,21)(16,24)(17,19)(20,22)(25,29,31,32)(26,27,33,34)(28,36,37,38)(30,39,40,35);(9,13)(10,12)(11,19)(14,21)(15,23)(16,22)(17,18)(20,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8300830);