Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,13)(11,15)(17,18)(19,24)(20,28)(21,26)(22,27)(23,30)(25,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,24)(20,29,31,21)(25,32,28,26);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,19,20,21,22,30,31,29)(18,24,28,26,27,23,25,32);(9,11,12,15)(10,14,13,16)(17,20,22,31)(18,25,27,28)(19,29,30,21)(23,32,24,26);(17,21,31,19,22,29,20,30)(18,26,25,24,27,32,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fghi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725916);