Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1292 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)(4,8)(5,7)(10,14)(12,15)(17,18,23,26)(19,25,27,32)(20,22,24,21)(28,31,30,29);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(18,24)(19,27)(20,26)(21,22)(25,30)(28,32);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,23,27)(18,25,26,32)(20,30,24,28)(21,31,22,29);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(17,20,23,24)(18,21,26,22)(19,28,27,30)(25,29,32,31);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,20,26,24)(19,29,27,31)(25,28,32,30)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7095753);