Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1761 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)(3,5)(4,6)(7,8)(9,10,15,21,16,22,14,20)(11,23,26,17,27,37,35,29)(12,28,30,38,31,25,36,18)(13,19,33,39,34,40,32,24);(1,2)(3,7)(4,6)(5,8)(9,11,16,27)(10,17,22,29)(12,24,31,39)(13,18,34,38)(14,35,15,26)(19,36,40,30)(20,23,21,37)(25,33,28,32);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,24)(13,29)(14,30)(15,36)(16,31)(17,34)(19,35)(20,28)(21,25)(22,38)(23,32)(26,40)(27,39)(33,37);(3,8)(5,7)(9,13,16,34)(10,19,22,40)(11,25,27,28)(12,29,31,17)(14,32,15,33)(18,35,38,26)(20,24,21,39)(23,36,37,30);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,27,35)(12,30,31,36)(13,32,34,33)(17,37,29,23)(18,28,38,25)(19,24,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970342);