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 | |
1832 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)(9,10,15,21,16,22,14,20)(11,23,25,17,26,35,33,36)(12,27,28,34,30,40,29,18)(13,19,32,39,24,37,31,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,33,16,26,15,25)(10,17,21,35,22,36,20,23)(12,13,28,32,30,24,29,31)(18,37,40,39,34,19,27,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,26)(14,28)(15,29)(16,30)(17,19)(20,27)(21,40)(22,34)(23,39)(25,32)(31,33)(35,38)(36,37);(3,8)(5,7)(9,13)(10,19)(11,12)(14,31)(15,32)(16,24)(17,34)(18,36)(20,38)(21,39)(22,37)(23,27)(25,28)(26,30)(29,33)(35,40);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,26,33)(12,28,30,29)(13,31,24,32)(17,35,36,23)(18,27,34,40)(19,38,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 6910289);