Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(1,3,4,8)(2,5,6,7)(10,14)(12,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(9,11,13,16)(10,15,14,12)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32);(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6528321);