Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| 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×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26)(27,33)(28,36)(29,31)(30,34)(32,38)(35,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24);(3,8)(5,7)(25,27,29,35)(26,30,31,39)(28,37,38,34)(32,40,36,33);(2,6)(5,7)(10,15)(11,18)(16,20)(19,24);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,31)(30,39)(32,36)(33,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6564732);