Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,13,20,14,19,28,35,15,18,29,34,12,17,26,33)(11,21,24,36,25,32,38,40,22,16,27,31,23,30,37,39);(5,6,7,8)(9,11)(10,16)(12,23)(13,27)(14,25)(15,22)(17,32)(18,21)(19,30)(20,36)(24,29)(26,38)(28,37)(31,34)(33,39)(35,40);(2,4)(10,17,18,19)(11,22)(16,30,21,32)(20,33,34,35)(23,25)(24,27)(31,39,36,40)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ci,f^b=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54021);