Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35190 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19,22,14)(12,17)(13,16)(15,18)(20,23,21,24);(9,11,19,18)(10,15,22,14)(12,20,24,17)(13,21,23,16);(2,5)(6,8)(9,12,15,13)(10,16,18,17)(11,20,22,21)(14,23,19,24);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,17,18,16)(11,21,22,20)(14,24,19,23)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311994, #72312002, #72312022, … |
| maximal normal subgroup of | #72311994, #72312002, #72312022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402802);