Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513086 |
| 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 | |
410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 410 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)(11,21)(12,18)(13,17)(14,29)(15,16)(19,26)(20,27)(22,30)(23,36)(24,34)(25,32)(28,33)(31,38)(35,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,16,18,21)(13,22,24,28)(14,23,25,27)(17,33,34,30)(19,31,35,37)(20,32,36,29)(26,39,40,38);(3,8)(5,7)(10,17)(11,15)(13,26)(14,27)(16,30)(18,34)(19,20)(21,33)(22,38)(23,25)(24,40)(28,39)(29,31)(32,37)(35,36);(2,6)(5,7)(10,18)(16,21)(17,34)(19,35)(20,36)(29,32)(30,33)(31,37)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=ce,f^c=fg,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420284);