Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
806 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,11,16,19,20,17,9,13,15,18,8,12,14,10);(1,2,3,4)(6,10)(7,14)(8,16)(11,12)(13,17)(15,20)(18,19);(7,15)(14,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72313772, #72313774, #72313790 |
| maximal normal subgroup of | #72313772, #72313774, #72313790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44353);