Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1590 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)(17,18,22,25)(19,23,20,28)(21,24,30,26)(27,32,29,31);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,22)(21,27)(24,32)(25,28)(26,31)(29,30);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(18,24)(19,20)(21,30)(23,31)(25,26)(28,32);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,23,25,28)(21,29,30,27)(24,32,26,31);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(18,25)(19,20)(24,26)(27,29)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6512562);