Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455641 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,18)(11,20)(12,17)(13,29,19,14)(16,31)(21,37,25,36)(22,24)(23,33)(26,30)(27,34)(28,35)(32,40,38,39);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,16,18,33)(12,25,27,21)(13,22,19,20)(14,31,29,23)(17,32,34,38)(26,39,28,40)(30,37,35,36);(3,7)(5,8)(9,12,14,26,15,27,29,28)(10,17,13,30,18,34,19,35)(11,21,23,39,24,25,31,40)(16,32,20,36,33,38,22,37);(9,13,14,10,15,19,29,18)(11,22,31,16,24,20,23,33)(12,17,26,35,27,34,28,30)(21,38,40,37,25,32,39,36)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=e,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381441);