Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,6)(5,7);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72216083, #72216108, #72216122, … |
| maximal normal subgroup of | #72216083, #72216108, #72216122, … |
| maximal quotient of | #72224736, #72264330, #72302495 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281443);