Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,20,24,21,22,28,29)(19,25,26,32,27,23,30,31);(1,3,6,2)(4,7,5,8);(9,11,13,12)(10,14,15,16);(1,4,6,5)(2,7,3,8)(18,22)(23,25)(24,29)(31,32);(1,5,6,4)(2,7,3,8)(9,12,13,11)(10,14,15,16)(17,19,20,26,21,27,28,30)(18,23,24,31,22,25,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=df140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314637);