Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,13,15)(11,14,12,16)(17,18,20,22)(19,21,23,24)(25,26,30,34)(27,32,36,40)(28,37,35,31)(29,38,39,33);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,35,30,36,39,28)(26,31,33,32,34,37,38,40);(3,7)(6,8)(9,12,13,11)(10,14,15,16)(19,23)(21,24)(27,28)(29,39)(31,40)(32,37)(33,38)(35,36);(1,3,4,7)(2,6,5,8)(25,28,29,27,30,35,39,36)(26,32,33,37,34,40,38,31)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378156);