Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36675 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,24,26,23,20,21,29)(19,27,28,22,30,25,31,32);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,23,30)(18,25,20,27)(21,31,24,28)(22,26,32,29);(9,11,12,16)(10,13,14,15)(17,20,24,29,23,18,21,26)(19,25,28,32,30,27,31,22);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,23,24)(18,26,20,29)(19,28,30,31)(22,27,32,25);(1,5,6,4)(2,7,3,8)(17,22,21,27,23,32,24,25)(18,19,29,31,20,30,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,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, 6338207);