Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,15,16,13)(19,21)(24,26)(28,31)(29,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,27)(19,28,21,31)(20,25,23,32)(24,29,26,30);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,23,30,22,21,20,29)(18,24,25,28,27,26,32,31);(10,14)(13,15)(17,20,22,23)(18,25,27,32)(19,29,21,30)(24,28,26,31);(17,21,23,29,22,19,20,30)(18,26,25,31,27,24,32,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=ghi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727986);