Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| 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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
970 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)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26)(27,35)(28,31)(29,39)(30,34)(32,37)(33,38)(36,40);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,28,30,37,38,36)(26,31,33,35,34,40,39,32);(3,7)(6,8)(10,15)(14,16)(19,23)(21,24)(27,36)(28,37)(29,38)(31,32)(33,39)(35,40);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(25,28,38,27,30,36,29,37)(26,32,39,40,34,35,33,31)Polycyclic presentation:
2^9|a^2=egi,b^2=fh,d^2=efi,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^c=de,e^c=ei,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, 378315);