Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #264203 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,11)(9,19)(10,12)(13,17)(14,16)(18,20)(21,22,26,30)(23,32,34,28)(24,35,33,36)(25,31,27,29);(5,7,16,18,10,17,9,8)(6,11,19,13,12,20,14,15)(21,23)(22,28)(24,27)(25,33)(26,34)(29,36)(30,32)(31,35);(1,2,3,4)(6,12)(11,20)(13,15)(14,19)(21,24,25,34,26,33,27,23)(22,28,29,35,30,32,31,36);(5,8,9,17,10,18,16,7)(6,13,14,11,12,15,19,20)(21,25,26,27)(22,29,30,31)(23,33,34,24)(28,36,32,35)Polycyclic presentation:
2^9|a^2=i,b^2=ehi,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 208071);