Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,19)(18,20);(5,7,10,8)(6,11,14,19)(9,16,18,17)(12,13,15,20);(5,8,10,7)(6,12,14,15)(9,16,18,17)(11,13,19,20);(1,3,4,2)(7,8)(11,19)(12,15)(16,17)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=g,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#33557, #72310467, #72310482, … |
| maximal normal subgroup of | C4×#33557, #72310467, #72310482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402942);