Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3055 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,14,18,15,17,12,13)(11,19,20,16,23,24,22,21);(5,7,8,6)(9,11)(10,16)(12,20)(13,24)(14,22)(15,23)(17,21)(18,19);(2,4)(9,12,15,14)(10,13,17,18)(11,20,23,22)(16,24,21,19);(9,13,12,17,15,18,14,10)(11,21,22,24,23,16,20,19);(10,17)(13,18)(16,21)(19,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#61066, #72284272 |
| maximal normal subgroup of | C4×#61066, #72284272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289179);