Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 730 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)(3,12,15,7)(4,10,17,8)(5,9)(6,11)(13,27,29,21)(14,26,31,22)(16,23,18,24)(19,25)(20,28,32,30);(1,3)(2,7)(4,14)(5,15)(6,13)(8,22)(9,12)(10,20)(11,21)(16,31)(17,28)(18,30)(19,29)(23,32)(24,26)(25,27);(2,8,9,23)(3,13)(4,16)(7,20,12,26)(10,25,24,11)(14,30)(15,29)(17,18)(21,22,27,32)(28,31)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=egi,f^c=fhi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72233265, #72233267, #72233269, … |
| maximal normal subgroup of | #72233265, #72233267, #72233269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42706);