Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,26,24,20)(22,25,31,32)(27,29,28,30);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,24)(18,20,23,26)(22,27,31,28)(25,30,32,29);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,23)(19,27)(20,29)(22,31)(24,28)(26,30);(1,4,5,3)(2,6,8,7)(17,20,21,26)(18,24,23,19)(22,30,31,29)(25,28,32,27);(19,27)(20,30)(22,31)(24,28)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,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, 6488770);