Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135518 |
| 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 | |
750 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 750 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)(3,13,16,23)(4,12)(5,10,17,8)(6,11)(7,14,22,29)(9,18)(15,27,31,21)(19,24,32,25)(20,28,26,30);(1,3,4,14)(2,7,12,23)(5,15,17,28)(6,16,18,29)(8,20,10,21)(9,13,11,22)(19,30,32,31)(24,26,25,27);(2,8,9,24)(5,19)(7,20,13,26)(10,11,25,12)(15,30)(17,32)(21,22,27,23)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=dh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eg,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233265, #72233266, #72233268, … |
| maximal normal subgroup of | #72233265, #72233266, #72233268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42715);