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 |
| 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 | |
407 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 112 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)(11,16)(12,17)(13,27)(14,29)(15,20)(18,23)(19,24)(21,32)(22,36)(25,30)(26,34)(28,40)(31,38)(33,37)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,20,29)(12,21,26,25)(13,22,28,24)(17,32,34,30)(18,31,35,33)(19,27,36,40)(23,38,39,37);(3,8)(5,7)(10,17)(11,14)(12,23)(13,24)(16,30)(18,19)(20,34)(21,37)(22,28)(25,38)(26,39)(27,33)(29,32)(31,40)(35,36);(2,6)(5,7)Polycyclic presentation:
2^9|b^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, 420290);