Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,12,16,10)(9,13)(14,18)(15,19,20,17);(1,2,3,4)(6,10,13,12)(11,17,18,19);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=be,e^a=eh,g^a=gi,h^a=hi,f^b=fhi,d^c=di,f^e=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225792, #72225802, #72232895, … |
| maximal normal subgroup of | #72225792, #72225802, #72232895, … |
| maximal quotient of | #72225786, #72232856, #72233041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400096);