Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35347 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2256 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)(3,4)(5,6,10,14,9,13,18,20)(7,15,16,17,12,11,19,8);(5,7,10,19)(6,11,20,17)(8,13,15,14)(9,12,18,16);(2,4)(5,8)(6,12)(7,13)(9,17)(10,15)(11,18)(14,19)(16,20);(6,13)(8,17)(11,15)(14,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233167, #72233184, #72233201, … |
| maximal normal subgroup of | #72233167, #72233184, #72233201, … |
| maximal quotient of | #72233159, #72233290, #72233440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401899);