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×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,16,30,14,20,15,21)(11,17,29,38,24,33,25,34)(12,19,28,35,13,18,26,37)(22,39,27,32,40,31,23,36);(1,3,4,7)(2,5,6,8)(9,11,14,24)(10,17,20,33)(12,23,13,27)(15,29,16,25)(18,32,19,36)(21,38,30,34)(22,28,40,26)(31,37,39,35);(2,6)(5,8)(9,12)(10,18)(11,22)(13,14)(15,28)(16,26)(17,31)(19,20)(21,37)(23,29)(24,40)(25,27)(30,35)(32,38)(33,39)(34,36);(3,7)(5,8)(9,13,14,12)(10,19)(11,23,24,27)(15,28,16,26)(17,32)(18,20)(21,37)(22,25,40,29)(30,35)(31,38)(33,36)(34,39)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385696);