Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,12,17,14,18,21,24,13,15,20,22,11,16,19,23);(1,3)(2,6)(4,8)(5,7)(10,15)(12,20)(16,18)(19,21);(1,4,6,5)(2,7,3,8)(9,11,13,14)(10,16,15,18)(12,19,20,21)(17,23,22,24);(10,15)(16,18)(17,22)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,e^b=egi,f^b=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388299);