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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1556 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,23)(19,21,25,31)(20,22,26,32)(27,30,29,28);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,24,25)(18,21,23,31)(20,28,26,30)(22,27,32,29);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20)(18,22)(19,27)(21,30)(23,32)(24,26)(25,29)(28,31);(1,4,5,3)(2,6,8,7)(17,21,24,31)(18,25,23,19)(20,29,26,27)(22,28,32,30);(17,22,24,32)(18,26,23,20)(19,28,25,30)(21,27,31,29)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,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, 6488711);