Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(10,15)(12,22)(13,24)(16,28)(17,32)(18,29)(20,35)(21,34)(23,26)(25,27)(30,39)(31,40)(33,38)(36,37);(2,6)(3,7)(9,10,14,18)(11,16,19,30)(12,23,25,32)(13,26,24,17)(15,27,29,22)(20,36,33,40)(21,37,34,31)(28,38,39,35);(1,3,5,6,4,7,8,2)(9,11)(10,16)(12,20)(13,21)(14,19)(15,28)(17,31)(18,30)(22,35)(23,36)(24,34)(25,33)(26,37)(27,38)(29,39)(32,40);(11,19)(16,30)(20,33)(21,34)(28,39)(31,37)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ch,e^b=eg,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391023);