Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,15,16,10)(9,12)(13,18)(14,20,19,17);(1,2,3,4)(7,13)(8,16)(11,18)(17,20);(6,10)(11,17)(12,15)(18,20);(5,7,9,13)(6,11,12,18)(8,14,16,19)(10,17,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231758, #72233476, #72234794, … |
| maximal normal subgroup of | #72231758, #72233476, #72234794, … |
| maximal quotient of | #72231753, #72231770, #72234760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401773);