Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,27)(20,28)(22,29)(23,30)(25,32)(26,31);(10,14)(13,16)(17,19,21,20)(18,23,24,26)(22,28,29,27)(25,31,32,30);(1,4,5,3)(2,6,7,8)(19,28)(20,27)(22,29)(23,31)(25,32)(26,30);(17,20,21,19)(18,23,24,26)(22,27,29,28)(25,31,32,30)Polycyclic presentation:
2^9|b^2=fgi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6487481);