Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C23 | #357 |
| 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 | |
161 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 161 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:
(1,2,3,5,4,6,7,8)(9,10)(11,22,24,53)(12,28,31,52)(13,35)(14,38)(15,41)(16,21)(17,23)(18,32,47,33)(19,30)(20,37)(25,58,26,59)(27,60,42,54)(29,49)(34,67,65,61)(36,70,39,46)(40,50)(43,68)(44,62)(45,57)(48,66,71,64)(51,56)(55,72)(63,69);(9,11,16,24)(10,17,21,45)(12,29,31,55)(13,36,14,39)(15,42,37,27)(18,46,47,70)(19,49,50,72)(20,51,41,44)(22,52,53,28)(23,32,57,33)(25,30,26,40)(34,68,65,63)(35,69,38,43)(48,59,71,58)(54,61,60,67)(56,66,62,64)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fhi,d^2=g,e^2=h,f^2=h,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=fhi,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, 1121);