Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | C2×C4⋊C16 | C22 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C2×C4⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 32 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:
(1,2,3,6,4,5,7,8)(9,10)(11,23,26,16)(12,29)(13,21)(14,20)(15,22)(17,24,36,38)(18,30)(19,25)(27,34,28,35)(31,37)(32,39,33,40);(2,5)(10,16)(11,24)(12,25)(13,14)(17,19)(18,36)(20,35)(21,34)(22,23)(26,38)(27,40)(28,39)(29,33)(30,31)(32,37)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=gh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1044);