Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1909 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,23)(19,30)(21,32)(22,37)(24,33)(25,40)(26,29)(27,36)(28,34)(31,38)(35,39);(1,3,5,4)(2,6,8,7)(9,11,13,23)(10,15,18,20)(12,25,27,39)(14,21,30,24)(16,33,19,32)(17,35,36,40)(22,28,38,29)(26,31,34,37);(10,16)(11,21)(14,30)(15,20)(17,34)(18,19)(22,25)(23,24)(26,36)(28,29)(31,37)(38,39);(1,3,5,4)(2,7,8,6)(9,10,13,18)(11,15,23,20)(12,26,27,34)(14,19,30,16)(17,28,36,29)(21,32,24,33)(22,35,38,40)(25,31,39,37);(1,4,5,3)(2,6,8,7)(9,12,13,27)(10,17,18,36)(11,22,23,38)(14,28,30,29)(15,31,20,37)(16,34,19,26)(21,25,24,39)(32,40,33,35)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485853);