Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
966 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)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,26,31,36)(27,34,38,40)(28,32,37,39)(29,33,30,35);(2,5)(6,8)(17,19)(18,21)(20,23)(22,24)(25,27,29,28,31,38,30,37)(26,32,35,34,36,39,33,40);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(19,23)(21,24)(26,33)(27,37)(28,38)(29,30)(32,39)(35,36);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(25,28,29,38,31,37,30,27)(26,34,35,39,36,40,33,32)Polycyclic presentation:
2^9|a^2=eg,b^2=hi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,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, 378207);