Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 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)(9,10)(11,15)(12,16)(13,14)(17,18,23,27)(19,21,29,24)(20,25,31,32)(22,30,28,26);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,25)(21,27)(22,30)(23,29)(26,28)(31,32);(10,14)(15,16)(17,20,23,31)(18,25,27,32)(19,28,29,22)(21,26,24,30);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,24)(18,19,27,29)(20,30,31,26)(22,32,28,25);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,22,23,28)(18,26,27,30)(19,20,29,31)(21,32,24,25)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6638127);