Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #39471 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,21)(22,29)(23,25)(26,30)(27,31)(28,32);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,26)(21,30)(22,27)(23,28)(25,32)(29,31);(10,14)(11,16)(17,20,21,18,23,29,22,25)(19,26,30,24,28,31,27,32);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,20,25,29)(19,27,28,30)(24,31,32,26);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6757229);