Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,11,9,12,8,14)(7,13,17,18,16,19,15,20)(21,22,25,28)(23,29,32,36)(24,33,30,27)(26,34,35,31);(1,3,4,2)(6,11)(7,15)(8,10)(12,14)(16,17)(18,20);(6,12)(11,14)(13,19)(18,20)(21,23,30,34,25,32,24,31)(22,26,33,36,28,35,27,29);(5,7,10,17,9,16,8,15)(6,13,11,18,12,19,14,20)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=fhi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138761);