Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #449 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,8,6,3,4,7)(9,10)(11,21)(12,24,26,32)(13,27)(14,16)(15,20)(17,22,35,40)(18,29)(19,23)(25,37,28,38)(30,36,39,34)(31,33);(1,3,6,2)(4,7,5,8)(9,11,13,14,15,23,29,31)(10,16,18,21,20,33,27,19)(12,22,28,30,26,40,25,39)(17,32,36,38,35,24,34,37);(1,2,6,3)(4,8,5,7)(9,12)(10,17)(11,22)(13,28)(14,30)(15,26)(16,32)(18,36)(19,37)(20,35)(21,38)(23,40)(24,33)(25,29)(27,34)(31,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=gh,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309613, #72309628, #72309632, … |
| maximal normal subgroup of | #72309613, #72309628, #72309632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44772);