Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,22)(13,17)(15,19)(16,27)(18,24)(20,28)(21,30)(23,26)(25,36)(29,31)(32,34)(33,39)(35,40)(37,38);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,23,26,22)(13,29,30,34)(15,19,24,18)(16,28,20,27)(17,32,21,31)(25,35,38,33)(36,39,37,40);(1,3,5,4)(2,7,8,6)(9,12,24,28,14,26,15,27)(10,16,19,23,11,20,18,22)(13,25,32,40,30,38,31,39)(17,33,29,37,21,35,34,36);(1,4,5,3)(2,6,8,7)(9,13,14,30)(10,17,11,21)(12,25,26,38)(15,31,24,32)(16,33,20,35)(18,34,19,29)(22,36,23,37)(27,39,28,40);(12,26)(16,20)(22,23)(25,38)(27,28)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351928);