Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| 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×#120 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,31,29,32,28,34)(27,35,38,40,37,33,36,39);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,30)(32,34)(37,38)(39,40);(2,6)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,32)(31,34)(33,35)(39,40);(12,20)(15,24)(17,19)(21,23)(25,27,30,38,29,37,28,36)(26,33,31,39,32,35,34,40);(10,16)(11,18)(15,24)(17,19)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6947992);