Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1402 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)(17,18,23,21)(19,28,30,20)(22,26,31,27)(24,25,32,29);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29,23,30,31,25)(18,24,27,28,21,32,26,20);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,31,32,23,28,22,24)(18,25,26,30,21,29,27,19);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,18)(19,24,30,32)(20,29,28,25)(22,26,31,27);(11,15)(14,16)(17,22,23,31)(18,26,21,27)(19,29,30,25)(20,32,28,24)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461186);