Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1512 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)(9,10)(11,14)(12,13)(15,16)(18,24)(25,32)(26,29)(27,28);(1,3,5,4)(2,6,8,7)(17,18)(19,29)(20,25)(21,28)(22,24)(23,27)(26,30)(31,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,25)(20,21)(22,30)(23,31)(24,32)(26,28)(27,29);(1,4,5,3)(2,6,8,7)(17,20,22,31)(18,26,24,29)(19,21,30,23)(25,28,32,27);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,28)(19,20,30,31)(25,29,32,26)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364110);