Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574075 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60457 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,21)(11,17,23,32)(12,18,25,31)(13,19,27,36)(14,28,15,20)(22,30,38,40)(24,33,37,39)(26,35,29,34);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,14)(13,24)(15,25)(16,23)(18,28)(19,33)(20,31)(21,32)(22,29)(26,38)(27,37)(30,34)(35,40)(36,39);(9,11,14,12,16,23,15,25)(10,18,20,17,21,31,28,32)(13,22,26,24,27,38,29,37)(19,33,34,30,36,39,35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,15)(13,22)(14,23)(16,25)(17,20)(19,30)(21,31)(24,26)(27,38)(28,32)(29,37)(33,35)(34,39)(36,40);(9,13)(10,19)(11,22)(12,24)(14,26)(15,29)(16,27)(17,30)(18,33)(20,34)(21,36)(23,38)(25,37)(28,35)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491539);