Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1811 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)(3,5)(4,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,26,17,27,34,33,35)(12,28,29,38,25,37,30,18)(13,19,32,39,24,36,31,40);(1,2)(3,7)(4,6)(5,8)(9,11,15,26,16,27,14,33)(10,17,21,23,22,35,20,34)(12,13,30,31,25,24,29,32)(18,36,28,40,38,19,37,39);(1,3)(2,5)(4,8)(6,7)(9,12,16,25)(10,18,22,38)(11,24,27,13)(14,29,15,30)(17,19,35,36)(20,37,21,28)(23,39,34,40)(26,32,33,31);(3,8)(5,7)(9,13)(10,19)(11,25)(12,27)(14,31)(15,32)(16,24)(17,18)(20,39)(21,40)(22,36)(23,37)(26,30)(28,34)(29,33)(35,38);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,27,33)(12,29,25,30)(13,31,24,32)(17,23,35,34)(18,37,38,28)(19,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^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=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910272);