Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884809 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1451 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)(3,6)(4,5)(7,8)(9,10,12,15)(11,16,13,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,7,4)(2,5,8,6)(9,11,12,13)(10,14,15,16)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(1,4,7,3)(2,5,8,6)(10,14)(11,13)(15,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(1,5,7,6)(2,3,8,4)(10,15)(14,16)(18,24)(22,25)(23,32)(30,31);(1,6,7,5)(2,3,8,4)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6894346);