Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,35)(29,31)(32,36)(33,38)(34,40)(37,39);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,33)(26,30,31,38)(28,34,36,39)(32,37,35,40);(1,2,4,3,5,6,8,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,31)(27,33)(32,35)(34,39);(11,19)(14,17)(18,24)(21,23)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,3,4,6,5,7,8,2)(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=df,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338880);