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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1234 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)(4,8)(5,6)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,32,37)(27,33,38,40)(28,34,30,39)(29,35,31,36);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,32,38)(26,33,37,40)(28,29,30,31)(34,35,39,36);(2,6)(7,8)(25,28,31,27,32,30,29,38)(26,34,36,33,37,39,35,40);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(25,29,32,31)(26,35,37,36)(27,30,38,28)(33,39,40,34);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,30,31,38,32,28,29,27)(26,34,36,33,37,39,35,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=gi,e^2=fgh,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750827);