Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25720 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 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)(11,15)(12,14)(13,16)(17,18,20,23)(19,25,27,32)(21,30,29,24)(22,31,26,28);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,23)(21,24)(22,27)(25,31)(28,32)(29,30);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(17,19,20,27)(18,22,23,26)(21,31,29,28)(24,32,30,25);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(18,23)(22,26)(24,30)(25,32);(1,6,7,5)(2,3,8,4)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900096);