Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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 | C2×#3460 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3705 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)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(2,5)(6,8)(12,19)(15,22)(17,23)(20,24);(1,3,4,7)(2,6,5,8);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=g,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279682);