Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
799 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)(11,18)(12,25)(13,28)(14,22)(15,21)(16,20)(17,24)(19,32)(23,37)(26,34)(27,35)(29,36)(30,40)(31,39)(33,38);(1,3,4,7)(2,5,6,8)(9,11,16,27,15,12,14,24)(10,17,22,25,21,35,20,18)(13,29,31,26,33,30,32,23)(19,36,38,37,39,40,28,34);(2,6)(5,8)(9,12,15,11)(10,18,21,25)(13,30,33,29)(14,24,16,27)(17,22,35,20)(19,37,39,34)(23,31,26,32)(28,40,38,36);(3,7)(5,8)(9,13,14,31,15,33,16,32)(10,19,20,38,21,39,22,28)(11,23,24,29,12,26,27,30)(17,34,18,36,35,37,25,40)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=i,d^2=e,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^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385705);