Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(26,30)(27,34)(32,37)(35,38);(1,2,5,4)(3,7,8,6)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,30)(31,39)(32,37)(33,40);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6763730);