Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,14,18,15,19,13,20)(11,21,23,32,25,16,24,36)(12,26,29,17,30,34,28,35)(22,38,27,37,40,33,39,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,33)(18,32)(19,21)(20,36)(22,29)(26,31)(28,40)(30,39)(34,37)(35,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,22)(13,29)(14,28)(15,30)(16,31)(18,26)(19,35)(20,34)(21,37)(23,39)(24,27)(25,40)(32,38)(33,36);(3,8)(5,6)(10,18)(11,23)(13,14)(16,21)(17,34)(19,20)(22,39)(24,25)(26,35)(27,40)(28,29)(33,38)Polycyclic presentation:
2^9|a^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385472);