Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(12,20)(15,21)(16,17)(18,24)(22,23)(25,26,29,32,28,30,35,37)(27,31,34,38,33,36,39,40);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24)(26,30)(31,36)(32,37)(38,40);(2,6)(5,8)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22)(25,27)(26,31)(28,33)(29,34)(30,36)(32,38)(35,39)(37,40);(10,16)(12,20)(14,17)(15,22)(18,24)(21,23)(27,33)(31,36)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dei,e^b=eh,d^c=di,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132264);