Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1845 |
| prime radical | C23 | #500 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(18,22)(19,26)(20,27)(23,28)(24,32)(25,31)(29,30);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16);(17,18,23,32,20,24,19,22,21,25,26,30,27,29,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=ehi,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72216239, #72216241, #72216244, … |
| maximal normal subgroup of | #72216239, #72216241, #72216244, … |
| maximal quotient of | #72216296 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42992);