Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234309 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,28,31,29,32,37,40)(27,33,34,38,35,39,36,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24);(2,5)(3,7)(25,27,29,35)(26,30,32,38)(28,36,37,34)(31,39,40,33);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23);(2,5)(6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7191092);