Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | #23386 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
764 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 764 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,27,35,12,17)(11,21,23,33,24,34,22,38,25,39,26,16,30,31,40,32);(1,2)(3,5)(4,6)(7,8)(9,11,14,30,15,25,27,24)(10,16,19,38,20,33,35,32)(12,26,13,22,28,23,29,40)(17,34,18,21,36,31,37,39);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44337);