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 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,37)(28,34)(29,32)(30,40)(31,36)(33,39)(35,38);(2,5)(6,8)(17,19)(18,21)(20,23)(22,24)(25,27,38,39,31,28,30,29)(26,32,40,34,36,33,35,37);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(19,23)(21,24)(25,28,31,27)(26,33,36,32)(29,38,39,30)(34,35,37,40);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(25,29,30,28,31,39,38,27)(26,34,35,32,36,37,40,33)Polycyclic presentation:
2^9|a^2=egi,b^2=hi,c^2=i,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, 378204);