Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1804 |
| prime radical | C23 | #416 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,22,28,24,30,44,27,23,29,45,58,43,57,21,41)(19,25,33,49,35,51,32,55,34,50,68,78,67,77,42,48)(20,36,39,52,40,56,72,60,37,59,73,26,46,53,38,54)(31,61,64,47,65,75,63,76,62,71,66,74,70,80,69,79);(17,19,24,35,23,34,43,67)(18,25,30,51,29,50,57,77)(20,31,46,70,37,62,40,65)(21,42,22,33,44,32,45,68)(26,47,60,79,52,74,54,76)(27,55,58,78,41,48,28,49)(36,71,53,75,59,61,56,80)(38,69,73,66,72,63,39,64);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,20,23,37)(18,26,29,52)(19,31,34,62)(21,39,44,73)(22,38,45,72)(24,46,43,40)(25,47,50,74)(27,56,41,53)(28,59,58,36)(30,60,57,54)(32,66,42,64)(33,69,68,63)(35,70,67,65)(48,75,55,80)(49,61,78,71)(51,79,77,76)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^b=ei,d^c=di,e^c=eh,f^c=fgh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41843);