Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,11)(8,13,19,12)(9,14)(10,15)(16,24)(17,22,29,25)(18,23)(20,27,32,26)(21,28)(30,34)(31,33,36,35);(1,2,4,3)(5,7,9,18,10,16,21,30)(6,11,14,23,15,24,28,34)(8,17,20,31,19,29,32,36)(12,25,26,35,13,22,27,33);(1,3,4,2)(6,12)(7,16)(8,19)(11,22)(13,15)(14,26)(18,30)(20,32)(23,33)(24,25)(27,28)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=ci,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54065);