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×#3285 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(12,20)(15,23)(16,21)(17,24)(18,22);(9,10,13,16)(11,17,19,23)(12,21,20,14)(15,22,24,18);(2,4);(9,11,13,19)(10,15,16,24)(12,22,20,18)(14,23,21,17);(5,7,8,6)(10,16)(14,21)(15,24)(17,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2943, #72286631, #72320495, … |
| maximal normal subgroup of | D4×#2943, #72286631, #72320495, … |
| maximal quotient of | #72287574, #72287738, #72309439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291009);