Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×C4⋊C8 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2,3,5,4,6,7,8);(2,6)(5,8);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=g,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72310101, M32×AGL(1,ℤ/8), #72310109, … |
| maximal normal subgroup of | #72310101, M32×AGL(1,ℤ/8), #72310109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390552);