Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C23 | #367 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 119 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 80 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,31)(20,37)(21,43)(22,46)(23,47)(24,30)(25,32)(26,40)(27,53)(28,38)(29,39)(33,66)(34,48)(35,69)(36,60)(41,56)(42,55)(44,73)(45,70)(49,68)(50,77)(51,65)(52,72)(54,71)(57,63)(58,61)(59,62)(64,74)(67,76)(75,79)(78,80);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,21,33,44,67,20,32,39,62,72,80,38,61,53,77,24,36,45,68,22,34,42,65,23,35,40,63,71,79,41,64)(18,25,27,48,54,66,52,60,47,74,37,58,46,57,43,59,30,51,56,76,28,49,26,31,29,50,55,75,73,78,70,69)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,h^a=hi,c^b=cg98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1596);