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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,26,22,25,32,21)(19,24,28,27,20,29,31,30);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,22)(21,30)(23,28)(25,29)(26,27)(31,32);(3,8)(5,7)(9,11,13,12)(10,14,15,16)(17,20,22,19)(18,24,25,29)(21,30,26,27)(23,31,32,28);(2,6)(5,7)(19,20)(24,29)(27,30)(28,31);(9,12,13,11)(10,14,15,16)(17,21,32,25,22,26,23,18)(19,27,31,24,20,30,28,29)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^b=di,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286113);