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 | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,19,21,24,30,26)(20,27,29,32,28,23,31,25);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,24,21,26,30,18)(20,25,29,27,28,32,31,23);(2,5)(6,8)(9,12,13,11)(10,14,15,16);(1,3,4,7)(2,6,5,8)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31);(20,28)(23,27)(25,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283873);