Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10,13,19,14,20,28,35,15,17,29,36,22,31,12,18)(11,21,25,32,26,37,23,38,24,39,27,16,30,33,40,34);(2,6)(3,4)(7,8)(9,11,22,26,15,24,14,30)(10,16,31,34,17,32,20,38)(12,27,29,40,28,25,13,23)(18,37,36,39,35,33,19,21);(2,6)(3,4)(7,8)(10,17)(12,28)(13,29)(16,32)(20,31)(23,40)(25,27)(34,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44180);