Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,24)(13,28)(14,18)(15,20)(17,21)(19,29)(22,32)(23,33)(25,34)(26,39)(27,40)(30,31)(35,38)(36,37);(1,3,5,4)(2,6,8,7)(9,11,14,22,15,23,29,13)(10,16,18,32,20,33,19,28)(12,25,26,37,27,38,30,21)(17,24,34,39,36,40,35,31);(9,12,15,27)(10,17,20,36)(11,21,23,37)(13,25,22,38)(14,30,29,26)(16,31,33,39)(18,35,19,34)(24,32,40,28);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=eg,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413655);