Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
704 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)(11,21)(12,26,29,32)(13,18)(14,19)(15,20)(16,24)(17,28,33,27)(22,36,37,30)(23,34)(25,31)(35,39,40,38);(1,3,4,8)(2,5,6,7)(9,11)(10,16,20,31)(12,22)(13,24)(14,25)(15,23)(17,30,33,36)(18,34,19,21)(26,35,32,40)(27,38)(28,39)(29,37);(2,6)(3,8)(9,12,13,27,15,29,14,28)(10,17,19,26,20,33,18,32)(11,22,24,38,23,37,25,39)(16,30,21,35,31,36,34,40);(2,6)(5,7)(10,18,20,19)(11,23)(16,21,31,34)(17,32,33,26)(22,37)(24,25)(30,35,36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=eg,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270346);