Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133822 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#2846 | C2 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
8367 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 8367 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,8,4,6,9,13)(5,7,10,14,11,12,15,16);(2,6)(3,9)(7,12)(10,15);(2,7)(6,12)Polycyclic presentation:
2^9|a^2=f,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,f^c=fh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209787, #72209860, #72209889, … |
| maximal normal subgroup of | #72209787, #72209860, #72209889, … |
| maximal quotient of | #72213758, C2×#134033, #72227594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41229);