Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,24,28,22,32)(19,29,21,31,20,26,30,25);(2,5)(6,8)(17,19,24,20)(18,25,28,31)(21,23,30,22)(26,32,29,27);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,30,24,19,22,21)(18,26,27,25,28,29,32,31);(17,21,23,20,24,30,22,19)(18,25,27,29,28,31,32,26);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,28,32)(19,21,20,30)(25,26,31,29)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372309);