Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
778 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)(3,7,8,5)(9,10,16,22)(11,18,25,33)(12,17,24,34)(13,27)(14,21,15,20)(19,30)(23,32)(26,36)(28,35)(29,38)(31,37)(39,40);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,22,34)(12,14,24,15)(13,28,31,26)(18,20,33,21)(19,36,38,35)(23,30,39,29)(27,40,37,32);(9,12,14,25,16,24,15,11)(10,18,20,34,22,33,21,17)(13,26,30,39,31,28,29,23)(19,35,27,40,38,36,37,32);(3,8)(5,7)(9,13)(10,19)(11,23)(12,26)(14,30)(15,29)(16,31)(17,32)(18,35)(20,27)(21,37)(22,38)(24,28)(25,39)(33,36)(34,40)Polycyclic presentation:
2^9|a^2=eg,b^2=efi,c^2=g,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,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351995);