Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| 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×#3026 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2151 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)(7,10)(8,16,18,12)(9,13)(11,15,20,17)(14,19);(1,2,3,4)(5,7)(6,10)(8,17)(9,14)(11,12)(13,19)(15,18)(16,20);(5,7)(6,11)(8,15)(9,14)(10,12)(13,20)(16,19)(17,18);(7,14)(10,19)(11,20)(15,17)Polycyclic presentation:
2^9|a^2=g,b^2=eg,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232683, #72232947, #72233051, … |
| maximal normal subgroup of | #72232683, #72232947, #72233051, … |
| maximal quotient of | #72232613, #72232926, #72233078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401778);