Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1919 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,20,23,36)(9,14)(10,27)(11,15)(12,19)(13,22,31,18)(16,21)(24,29,25,34)(26,30)(28,32,35,33);(5,7,11,10,9,19,21,26)(6,12,15,30,14,17,16,27)(8,18,24,28,23,22,25,35)(13,29,32,36,31,34,33,20);(5,8,21,25,9,23,11,24)(6,13,15,32,14,31,16,33)(7,18,26,35,19,22,10,28)(12,29,30,36,17,34,27,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54096);