Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #1786 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,7)(4,8)(9,10,14,20,12,21,28,18,15,19,31,36,29,17,13,30)(11,22,26,34,27,35,40,38,25,37,39,16,23,33,24,32);(1,2,5,3)(4,7,8,6)(9,11,15,25)(10,16,19,34)(12,23,29,27)(13,26,28,39)(14,24,31,40)(17,32,21,38)(18,35,30,33)(20,37,36,22);(1,3,5,2)(4,6,8,7)(9,12,15,29)(10,17,19,21)(11,23,25,27)(13,31,28,14)(16,32,34,38)(18,36,30,20)(22,33,37,35)(24,26,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=egi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39140);