Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| socle | C2 | #9492 |
| prime radical | #2010 | C4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,3,6,11,8,16,15,5,9,10,14,4,13,12,7);(2,6,8,15,9,14,13,7);(3,10)(12,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231251, #72240973, C2×#139694, … |
| maximal normal subgroup of | #72231251, #72240973, C2×#139694, … |
| maximal quotient of | C2×#139694, #72309245, #72338350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46890);