Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4626 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)(3,6)(4,5)(7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(2,5)(6,8);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,3,4,7)(2,6,5,8);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,h^c=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224727, #72224729, #72224732, … |
| maximal normal subgroup of | #72224727, #72224729, #72224732, … |
| maximal quotient of | #72239312, #72239979, #72257827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278133);