Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62832 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(3,8)(5,7)(10,15)(11,17)(12,19)(20,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24);(11,17)(14,21)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431908);