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 | |
526 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18)(19,23,24,21)(20,22)(25,26,27,29,28,30,31,32);(2,5,7,6)(9,10,11,14,12,13,15,16)(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=g,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 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6);