Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2065 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,15,26,32,12,17,25,34,9,27,8,14)(7,18,22,29,23,31,21,33,20,35,24,13,19,30,36,28);(1,2,3,4)(7,19,20,23)(13,28,29,33)(18,31,35,30)(21,22,36,24);(5,7)(6,13)(8,24)(9,23)(10,21)(11,19)(12,20)(14,31)(15,33)(16,18)(17,29)(22,26)(25,36)(27,28)(30,32)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^c=di,e^c=eh,f^c=fi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55398);