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 | #35527 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2340 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)(5,6,8,13,9,12,10,17)(7,14,16,20,15,11,18,19);(5,6,10,13)(7,11,18,19)(8,17,9,12)(14,16,20,15);(2,4)(7,15)(11,14)(16,18)(19,20);(5,7,9,15)(6,11,12,14)(8,16,10,18)(13,19,17,20)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,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 | #72234776, #72234782, #72234834, … |
| maximal normal subgroup of | #72234776, #72234782, #72234834, … |
| maximal quotient of | #72234778, #72240233, #72260404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401901);