Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3199 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)(11,16)(12,24,26,34)(13,18)(14,20)(15,19)(17,25,32,28)(21,35,37,39)(22,30)(23,31)(27,33)(29,38,40,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,27)(15,23)(17,29)(18,30)(19,31)(20,33)(24,39)(25,36)(26,37)(28,38)(32,40)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,24)(19,32)(20,34)(22,36)(23,37)(27,38)(30,39)(31,40)(33,35);(3,8)(5,7)(9,13,15,14)(10,18,19,20)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,24,32,34)(21,36,37,38)(29,39,40,35);(2,6)(5,7)(10,19)(16,31)(17,32)(18,20)(24,34)(29,40)(30,33)(35,39)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407197);