Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1271 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1271 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,14,12,13,15,16);(10,13)(11,15);(2,5,6,7)(3,8)(10,13)Polycyclic presentation:
2^9|a^2=f,c^2=dh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=egi,f^c=fh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72232734, #72232774, #72232775, … |
| maximal normal subgroup of | #72232734, #72232774, #72232775, … |
| maximal quotient of | #72338845, #72339046 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42767);