Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196338 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,26,32)(11,15,21,29,20,28,36,39)(12,22,23,33,24,16,25,31)(19,27,35,38,34,30,37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,30)(17,28)(18,29)(22,27)(23,35)(24,34)(25,37)(26,36)(31,40)(32,39)(33,38);(9,12,14,24)(10,16,17,22)(11,19,20,34)(13,25,26,23)(15,27,28,30)(18,33,32,31)(21,37,36,35)(29,40,39,38);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266904);