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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1501 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)(17,18,23,24)(19,28,30,25)(20,22,27,32)(21,26,29,31);(1,3,5,4)(2,6,7,8)(17,19,23,30)(18,25,24,28)(20,31,27,26)(21,22,29,32);(9,11,12,16)(10,13,14,15)(17,20)(18,22)(19,29)(21,30)(23,27)(24,32)(25,26)(28,31);(1,4,5,3)(2,6,7,8)(17,21)(18,26)(19,22)(20,25)(23,29)(24,31)(27,28)(30,32);(17,22,23,32)(18,27,24,20)(19,26,30,31)(21,28,29,25)Polycyclic presentation:
2^9|a^2=gi,b^2=f,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=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, 6493389);