Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1662 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)(7,17,20,30)(8,13)(9,14)(10,16)(11,15)(12,19,26,23)(18,32,34,36)(21,28)(22,29)(24,31)(25,35,27,33);(5,7,10,19,11,20,9,23)(6,12,16,17,15,26,14,30)(8,18,21,35,22,34,24,33)(13,27,28,36,29,25,31,32);(1,2,3,4)(5,8)(6,13)(7,18)(9,24)(10,21)(11,22)(12,25)(14,31)(15,29)(16,28)(17,32)(19,35)(20,34)(23,33)(26,27)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31219);