Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | #3002 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1847 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,28)(13,20)(14,15)(16,33)(17,21)(18,26)(19,31)(23,38)(24,29)(25,34)(27,37)(30,32)(35,36)(39,40);(1,3,5,4)(2,6,8,7)(9,11)(10,18)(12,29)(13,24)(14,22)(15,26)(16,27)(17,34)(19,38)(20,36)(21,37)(23,32)(25,33)(28,35)(30,39)(31,40);(1,3,5,4)(2,7,8,6)(9,12,15,20)(10,19,14,30)(11,23,26,40)(13,33,28,21)(16,32,17,31)(18,35,22,24)(25,39,37,38)(27,36,34,29);(1,3,5,4)(2,7,8,6)(9,13)(10,20)(11,24)(12,14)(15,28)(16,19)(17,30)(18,36)(21,32)(22,29)(23,37)(25,40)(26,35)(27,38)(31,33)(34,39);(1,4,5,3)(2,6,8,7)(9,14)(10,15)(11,25)(12,30)(13,31)(16,33)(17,21)(18,27)(19,20)(22,34)(23,39)(24,29)(26,37)(28,32)(35,36)(38,40)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,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, 6486220);