Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(18,23)(20,21)(24,31)(25,30)(26,32)(27,29);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,26,28,24)(20,23,21,30)(27,31,29,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,18)(19,24)(20,23)(21,30)(22,25)(26,28)(27,32)(29,31);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,19,22,28)(18,24,25,26)(20,29,21,27)(23,31,30,32);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,20,22,21)(18,23,25,30)(19,27,28,29)(24,32,26,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fgi,d^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6471317);