Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513092 |
| 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 | C2×#2215 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,15)(12,16,20,28)(13,26,21,38)(14,27,22,19)(17,33,29,23)(18,35,30,24)(25,36,37,32)(31,40,34,39);(1,3)(2,5)(4,8)(6,7)(10,15)(16,28)(17,29)(18,30)(19,27)(26,38)(31,34)(32,36);(3,8)(5,7)(10,16)(12,23)(13,24)(14,22)(15,28)(17,18)(19,36)(20,33)(21,35)(25,40)(26,31)(27,32)(29,30)(34,38)(37,39);(2,6)(5,7)(10,15)(16,28)(17,29)(18,30)(19,27)(26,38)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=cei,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420289);