Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500385 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1545 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)(10,14)(11,16)(12,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(2,5)(6,8)(9,10,15,16,13,11,12,14)(17,19,21,25)(18,22,23,29)(20,26,27,31)(24,28,30,32);(1,3,4,7)(2,6,5,8)(9,11,12,16,13,10,15,14)(18,23)(19,25)(20,27)(28,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,28,29,32);(10,11)(14,16)(19,25)(22,29)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433746);