Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1331 |
| prime radical | C23 | #359 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,20,34,50,38,54,67,72,64,70,33,49,16,22,36,52,39,55,32,48,15,21,35,51,31,47,12,18)(11,23,25,44,27,57,62,71,40,68,30,17,37,41,61,43,29,59,63,56,26,46,24,53,28,58,60,45,65,69,66,42);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,30)(13,24)(14,37)(15,40)(16,29)(18,46)(19,23)(20,53)(21,56)(22,45)(25,34)(26,31)(27,64)(28,38)(32,63)(33,60)(35,66)(36,62)(39,65)(41,50)(42,47)(43,70)(44,54)(48,69)(49,57)(51,68)(52,59)(55,71)(58,72)(61,67)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fh,g^b=ghi,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1129);