Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,37)(28,32)(29,35)(30,33)(31,36)(34,39)(38,40);(2,5)(6,8)(17,19)(18,21)(20,23)(22,24)(25,27,29,28,31,39,30,38)(26,32,33,37,36,40,35,34);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(19,23)(21,24)(26,33)(27,38)(28,39)(29,30)(34,37)(35,36);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(25,28,30,27,31,38,29,39)(26,34,35,40,36,37,33,32)Polycyclic presentation:
2^9|a^2=fg,b^2=h,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, 378248);