Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #127312 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C4×M16 | C23 |
| socle | C2 | C2×#1288 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,16,18,24)(5,10)(6,11)(7,14,20,28)(8,19,23,31)(9,17)(13,25)(15,27)(21,32,22,30)(26,29);(1,3)(2,7,9,21,11,20,10,22)(4,14)(5,15)(6,13)(8,12,24,26,23,25,16,27)(17,29)(18,28)(19,30)(31,32);(3,13)(7,20)(12,25)(14,28)(15,29)(21,22)(26,27)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ghi,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286532, #72286539, #72286573, … |
| maximal normal subgroup of | #72286532, #72286539, #72286573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34508);