Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135024 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #476 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,10,13,15,11,16,12,7)(18,24)(19,25,29,27)(20,23)(21,26)(28,30,31,32);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(17,18,20,24,22,26,23,21)(19,27,28,30,29,25,31,32);(3,11)(7,15)(10,16)(12,13)(17,19,22,29)(18,25,26,27)(20,28,23,31)(21,30,24,32)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=i,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285431, #72285438, #72285579, … |
| maximal normal subgroup of | #72285431, #72285438, #72285579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42263);