Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18,21,20)(19,22,23,24)(25,26,31,35)(27,36,28,39)(29,34,30,33)(32,38,40,37);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,21,23)(18,22,20,24)(25,27,30,38,31,28,29,37)(26,32,34,39,35,40,33,36);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,20,21,18)(19,22,23,24)(26,33)(27,37)(28,38)(29,30)(32,40)(34,35);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(25,28,30,37,31,27,29,38)(26,32,34,39,35,40,33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,c^2=eg,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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382059);