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 | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
440 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)(9,10,14,20,16,19,15,29)(11,21,23,38,24,37,30,31)(12,18,13,27,26,35,25,36)(17,28,33,39,32,40,34,22);(9,11,16,24)(10,17,19,32)(12,22,26,39)(13,28,25,40)(14,23,15,30)(18,31,35,38)(20,33,29,34)(21,36,37,27);(1,3,4,8,5,2,6,7)(9,12,15,25,16,26,14,13)(10,18,29,36,19,35,20,27)(11,22,30,40,24,39,23,28)(17,31,34,37,32,38,33,21);(2,3)(7,8)(10,19)(17,32)(18,35)(20,29)(21,37)(27,36)(31,38)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,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=dhi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391278);