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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2081 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,26)(13,20)(14,33)(15,34)(16,22)(17,23)(19,28)(21,24)(25,38)(27,35)(29,40)(30,36)(31,39)(32,37);(2,5)(6,8)(9,11,17,25)(10,18,23,38)(12,27,30,31)(13,16,32,15)(14,29,24,28)(19,33,40,21)(20,22,37,34)(26,35,36,39);(9,12,17,30)(10,19,23,40)(11,24,25,14)(13,31,32,27)(15,29,16,28)(18,35,38,39)(20,21,37,33)(22,36,34,26);(1,3,4,7)(2,6,5,8)(9,13)(10,20)(11,16)(12,14)(15,25)(17,32)(18,22)(19,39)(21,36)(23,37)(24,30)(26,33)(27,29)(28,31)(34,38)(35,40);(9,14)(10,21)(11,12)(13,28)(15,31)(16,27)(17,24)(18,36)(19,37)(20,40)(22,39)(23,33)(25,30)(26,38)(29,32)(34,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6482652);