Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| 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 | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
434 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)(9,10,14,18,15,20,31,19)(11,21,24,34,26,39,25,40)(12,27,13,29,28,17,32,37)(16,33,35,22,36,30,38,23);(9,11,15,26)(10,16,20,36)(12,22,28,23)(13,30,32,33)(14,24,31,25)(17,34,27,40)(18,35,19,38)(21,37,39,29);(1,3,4,8,5,2,6,7)(9,12,14,13,15,28,31,32)(10,17,18,37,20,27,19,29)(11,22,24,30,26,23,25,33)(16,34,35,39,36,40,38,21);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391287);