Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,22)(12,18)(13,29)(14,19)(15,32)(16,21)(17,24)(20,31)(23,34)(25,33)(26,40)(27,36)(28,37)(30,35)(38,39);(9,11,14,25,15,26,31,24,16,27,28,39,12,23,13,30)(10,17,20,34,18,33,19,36,21,35,29,40,32,38,37,22);(9,12,16,15)(10,18,21,32)(11,23,27,26)(13,28,31,14)(17,33,35,38)(19,29,37,20)(22,34,36,40)(24,25,30,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33860);