Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500482 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,36)(23,32)(25,31)(28,33)(29,34)(30,39)(35,38)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,13,24,15,25,14,23)(10,16,19,21,20,32,18,31)(12,22,27,38,28,39,29,37)(17,30,26,35,34,36,33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(3,8)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,39,38)(30,35,36,40);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407680);