Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,14,19)(11,20,23,39)(12,17,15,29)(13,25,27,37)(16,31,33,22)(18,24,36,28)(21,38,30,35)(26,32,40,34);(9,11,14,23)(10,16,19,33)(12,21,15,30)(13,26,27,40)(17,32,29,34)(18,35,36,38)(20,37,39,25)(22,24,31,28);(1,3,4,8,5,2,6,7);(2,3)(7,8)(9,12,14,15)(10,17,19,29)(11,21,23,30)(13,24,27,28)(16,32,33,34)(18,25,36,37)(20,38,39,35)(22,26,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dh110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391264);