Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143938 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2065 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,3,6,4,7,8,5)(9,10,17,23)(11,24)(12,25)(13,21,33,40)(14,20,15,28)(16,36,35,22)(18,38)(19,31)(26,32)(27,29)(30,39)(34,37);(9,11,13,25,16,19,14,34,17,27,33,38,35,39,15,26)(10,18,20,24,22,37,21,30,23,12,28,29,36,32,40,31);(2,5)(3,8)(6,7)(9,12)(10,19)(11,22)(13,31)(14,29)(15,24)(16,37)(17,18)(20,38)(21,34)(23,39)(25,28)(26,40)(27,36)(30,33)(32,35)Polycyclic presentation:
2^9|a^2=de,b^2=d,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fg,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eh,f^c=fg,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53104);