Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 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)(18,24)(20,25)(27,31)(28,30);(1,3,5,4)(2,6,7,8)(17,18,22,24)(19,28,29,30)(20,26,25,32)(21,27,23,31);(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,25,24,20)(21,26,23,32)(27,30,31,28);(1,4,5,3)(2,6,7,8)(17,20)(18,26)(19,27)(21,30)(22,25)(23,28)(24,32)(29,31);(17,21)(19,26)(22,23)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493244);