Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,19)(12,21)(13,25)(16,29)(17,32)(18,34)(20,24)(22,38)(23,28)(26,33)(27,30)(31,39)(35,36)(37,40);(2,5)(6,8);(9,10,14,16)(11,20,23,37)(12,24,26,40)(13,27,25,18)(15,28,29,19)(17,33,31,21)(22,39,35,32)(30,36,34,38);(1,3,4,7)(2,6,5,8)(10,16)(15,29)(17,31)(18,27)(20,37)(24,40)(30,34)(32,39)Polycyclic presentation:
2^9|c^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^c=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234339, #72234341, #72234354, … |
| maximal normal subgroup of | #72234339, #72234341, #72234354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420202);