Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,5,8,6,10,3,13)(4,9,17,22,16,26,30,28)(7,12,19,23,21,25,14,29)(11,20,24,32,27,15,31,18);(1,3)(2,8)(4,15)(5,6)(7,14)(9,24)(10,13)(11,22)(12,23)(16,20)(17,32)(18,30)(19,21)(25,29)(26,31)(27,28);(2,9,25,27)(3,14)(4,16)(5,19)(8,22,29,31)(10,26,12,11)(13,28,23,24)(15,17)(18,32)(20,30)Polycyclic presentation:
2^9|a^2=f,c^2=dh,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=eg,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233221, #72233266, #72233268, … |
| maximal normal subgroup of | #72233221, #72233266, #72233268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42765);