Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2714 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,26,23,28,21,32)(19,29,20,31,30,22,27,25);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,24)(21,27)(23,30)(25,26)(28,29)(31,32);(10,14)(13,16)(17,20)(18,25)(19,21)(22,32)(23,27)(24,30)(26,29)(28,31);(1,4,6,5)(2,7,3,8)(17,21,23,24)(18,26,28,32)(19,20,30,27)(22,31,29,25);(1,5,6,4)(2,7,3,8)(17,22,21,31,23,29,24,25)(18,27,32,30,28,20,26,19)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310124);