Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
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×C22⋊C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2)(3,6)(4,8)(5,7);(1,3,5,4)(2,6,7,8);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72310098, M32×AGL(1,ℤ/8), #72310107, … |
| maximal normal subgroup of | #72310098, M32×AGL(1,ℤ/8), #72310107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388356);