Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | |
1785 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)(3,6)(4,5)(7,8)(9,10,15,14)(11,21,25,17)(12,27,31,36)(13,20,16,19)(18,32,30,29)(22,35,38,40)(23,24,26,34)(28,37,39,33);(2,5)(3,7)(9,11,15,25)(10,17,14,21)(12,28,31,39)(13,23,16,26)(18,35,30,40)(19,24,20,34)(22,29,38,32)(27,33,36,37);(9,12,16,29,15,31,13,32)(10,18,20,27,14,30,19,36)(11,22,26,28,25,38,23,39)(17,33,34,40,21,37,24,35);(2,5)(6,8)(9,13,15,16)(10,19,14,20)(11,23,25,26)(12,29,31,32)(17,24,21,34)(18,27,30,36)(22,28,38,39)(33,40,37,35);(1,3,4,7)(2,6,5,8)(9,14,15,10)(11,24,25,34)(12,30,31,18)(13,20,16,19)(17,23,21,26)(22,35,38,40)(27,29,36,32)(28,33,39,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592715);