Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 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)(3,4)(9,10,14,19,15,20,31,21)(11,22,25,38,26,40,27,34)(12,17,13,29,28,18,32,37)(16,33,35,23,36,30,39,24);(5,6,7,8)(9,11,15,26)(10,16,20,36)(12,23,28,24)(13,30,32,33)(14,25,31,27)(17,34,18,38)(19,35,21,39)(22,37,40,29);(2,4)(9,12,14,13,15,28,31,32)(10,17,19,29,20,18,21,37)(11,23,25,30,26,24,27,33)(16,34,35,22,36,38,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dh,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54409);