Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1338 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)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,33,31,34,29,39)(27,35,37,28,36,38,40,32);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,36)(26,32,34,28)(29,37,30,40)(33,38,39,35);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28,31,32)(26,27,34,36)(29,35,30,38)(33,37,39,40);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,36)(28,32)(35,38)(37,40);(10,16)(14,21)(15,24)(22,23)(25,29,31,30)(26,33,34,39)(27,37,36,40)(28,35,32,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392027);