Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | #2433 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,27,30,24)(20,25,32,29)(21,26,28,31);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,31)(21,29)(22,30)(23,27)(25,28)(26,32);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,22,32)(18,25,23,29)(19,28,30,21)(24,26,27,31);(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,29)(20,27)(22,28)(23,31)(24,32)(25,30);(1,4,5,3)(2,6,8,7)(18,23)(19,27)(20,32)(21,31)(24,30)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497945);