Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(3,7)(5,8)(9,10,15,19)(11,21)(12,16)(13,27,30,36)(14,18,25,20)(17,31,37,29)(22,39)(23,34)(24,33)(26,38)(28,32)(35,40);(1,2,4,6)(3,8,7,5)(9,11,15,24)(10,16,19,34)(12,25,23,14)(13,22,30,28)(17,32,37,39)(18,33,20,21)(26,29,40,31)(27,35,36,38);(1,3,4,7)(2,5,6,8)(9,12,14,11,15,23,25,24)(10,16,20,33,19,34,18,21)(13,28,29,40,30,22,31,26)(17,35,27,32,37,38,36,39);(9,13,15,30)(10,17,19,37)(11,22,24,28)(12,26,23,40)(14,31,25,29)(16,32,34,39)(18,27,20,36)(21,38,33,35)Polycyclic presentation:
2^9|a^2=efg,b^2=gh,c^2=g,d^2=i,e^2=i,g^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381944);