Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134947 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8642 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,11,10)(3,13,15,31)(4,18)(5,20)(6,19)(9,25)(12,30,23,24)(14,17,21,16)(22,32)(26,29);(1,2,4,8,6,12,18,24,7,11,5,10,19,23,20,30)(3,9,14,22,16,27,13,26,15,25,21,29,17,28,31,32);(1,3)(2,9)(4,14)(5,21)(6,16)(7,15)(8,22)(10,29)(11,25)(12,27)(13,18)(17,19)(20,31)(23,28)(24,26)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=ehi,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214153, #72214178, … |
| maximal normal subgroup of | #72214066, #72214153, #72214178, … |
| maximal quotient of | #72216393, C3×#134947, #72227256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42143);