Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(9,10,15,21,14,20,16,30)(11,22,25,32,24,38,31,37)(12,19,13,28,26,18,27,36)(17,29,34,39,33,40,35,23);(5,6,7,8)(9,11,14,24)(10,17,20,33)(12,23,26,39)(13,29,27,40)(15,25,16,31)(18,32,19,37)(21,34,30,35)(22,36,38,28);(1,2,3,4)(9,12,16,27,14,26,15,13)(10,18,30,28,20,19,21,36)(11,23,31,40,24,39,25,29)(17,32,35,22,33,37,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,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, 54641);