Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24);(10,14)(11,15);(10,14)(13,16)(18,21)(19,24)(22,23);(1,3,4,8)(2,6,5,7)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282132, #72282345, #72284973, … |
| maximal normal subgroup of | #72282132, #72282345, #72284973, … |
| maximal quotient of | #72305594, #72305683, #72315835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434259);