Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(18,21)(22,23)(25,26)(27,30)(28,31)(29,33)(32,36)(34,37)(35,39)(38,40);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18,20,21)(19,23,24,22)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,3)(2,6)(4,8)(5,7)(11,15)(13,16)(18,21)(19,24);(3,8)(6,7)(17,19,20,24)(18,22,21,23)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=eg,b^2=efi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386493);