Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145015 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,4,7)(3,5,8,6)(9,10)(11,14)(12,26)(13,22)(15,19)(16,23)(17,20)(18,29)(21,32)(24,31)(25,34)(27,40)(28,38)(30,39)(33,36)(35,37);(9,11,16,25)(10,17,23,32)(12,27,31,36)(13,21,15,20)(14,19,34,22)(18,28,37,30)(24,35,26,29)(33,38,40,39);(2,5)(3,8)(6,7)(9,12,13,28,16,31,15,30)(10,18,19,36,23,37,22,27)(11,24,20,38,25,26,21,39)(14,33,32,29,34,40,17,35)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=egi,f^c=fh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52301);