Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93766 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1333 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 158 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)(9,10,13,19,15,22,38,20,16,24,39,21,41,23,14,40)(11,25,27,55,29,57,65,56,30,59,66,48,42,58,28,67)(12,31,33,18,36,49,35,50,37,68,34,52,71,54,60,51)(17,32,43,64,45,62,53,69,47,70,72,26,46,61,44,63);(1,3)(2,5)(4,8)(6,7)(9,11,16,30)(10,17,24,47)(12,32,37,70)(13,28,39,65)(14,27,38,66)(15,42,41,29)(18,48,52,55)(19,53,21,44)(20,43,40,72)(22,46,23,45)(25,54,59,49)(26,60,64,35)(31,57,68,58)(33,63,34,69)(36,61,71,62)(50,56,51,67)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=g,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1119);