Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #95 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×C43 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C43 ▷ C24 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6)(3,4)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23,24,21)(20,22)(25,26,27,29,28,30,31,32);(1,2,4,7,5,8,3,6)(10,13)(11,16)(14,15)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=fgh,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh104 conjugacy classes — view the full character table.
| maximal quotient of | #72237309, #72237313 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9);