Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C22×D16 | C4 |
| 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 | |
179 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 179 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21,24,15)(12,26)(13,29)(14,20)(16,22,34,37)(17,27)(18,23)(19,30)(25,33,40,36)(28,35)(31,38,32,39);(10,15)(11,22)(12,23)(13,30)(16,18)(17,34)(19,36)(20,21)(24,37)(25,39)(26,32)(27,28)(29,33)(31,35)(38,40)Polycyclic presentation:
2^9|a^2=d,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.
| maximal subgroup of | #72216293 |
| maximal normal subgroup of | #72216293 |
| maximal quotient of | #72227205, #72227269 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1120);