Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1560 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)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,24,27,23)(21,25,28,30)(26,31,29,32);(1,3,5,4)(2,6,7,8)(18,22)(19,26)(21,28)(23,31)(24,32)(27,29);(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,23,22,24)(21,29,28,26)(25,31,30,32);(1,4,5,3)(2,6,7,8)(19,26)(21,28)(23,32)(24,31)(25,30)(27,29);(17,19,20,27)(18,24,22,23)(21,29,28,26)(25,32,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493229);