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 | #50359 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(1,2,4,6)(3,5,8,7)(10,12)(15,17)(16,20)(23,24);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,12,14,10)(11,15,19,17)(13,20,22,16)(18,23,21,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,40)(35,37)(38,39);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358799);