Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2094 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 266 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:
(1,2,6,8,5,3,4,7)(9,10)(11,19,22,33)(12,15)(13,17)(14,18)(16,21,28,25)(20,37,40,36)(23,29)(24,30)(26,34)(27,32,39,31)(35,38);(1,3,5,2)(4,7,6,8)(10,15)(11,20,22,40)(13,23)(14,26)(16,31,28,32)(18,35)(19,37,33,36)(21,27,25,39)(24,38)(30,34);(1,2,5,3)(4,8,6,7)(9,11)(10,16)(12,22)(13,25)(14,27)(15,28)(17,33)(18,36)(19,29)(20,38)(21,23)(24,39)(26,40)(30,37)(31,34)(32,35)Polycyclic presentation:
2^9|a^2=dgi,b^2=deh,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53475);