Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884873 |
| 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 | #36374 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1565 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)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,29,33)(27,34,30,31)(28,36,37,40)(32,38,39,35);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,36,39,40);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,29,37)(26,32,33,39)(27,35,30,38)(31,36,34,40);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,33)(28,37)(31,34)(35,38);(10,14)(15,21)(16,22)(23,24)(28,37)(32,39)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6860579);