Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,31,37)(27,38,39,33)(28,36,30,34)(29,40,32,35);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,31,39)(26,33,37,38)(28,32,30,29)(34,35,36,40);(2,5)(6,8)(25,28,32,39,31,30,29,27)(26,34,35,38,37,36,40,33);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,31,32)(26,35,37,40)(27,28,39,30)(33,36,38,34);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,30,29,39,31,28,32,27)(26,36,40,38,37,34,35,33)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6733601);