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)(17,18,22,28,23,29,44,58,24,30,45,59,43,57,21,27)(19,25,33,49,34,50,32,48,35,51,67,78,66,77,41,55)(20,36,39,52,40,56,38,60,37,71,42,26,46,53,72,54)(31,61,64,47,65,75,68,76,62,70,73,74,69,80,63,79);(17,19,23,34,24,35,43,66)(18,25,29,50,30,51,57,77)(20,31,46,69,37,62,40,65)(21,41,22,33,44,32,45,67)(26,47,60,79,52,74,54,76)(27,55,28,49,58,48,59,78)(36,70,53,75,71,61,56,80)(38,68,39,64,72,63,42,73);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,20)(18,26)(19,31)(21,42)(22,38)(23,46)(24,37)(25,47)(27,56)(28,36)(29,60)(30,52)(32,64)(33,68)(34,69)(35,62)(39,44)(40,43)(41,73)(45,72)(48,75)(49,70)(50,79)(51,74)(53,58)(54,57)(55,80)(59,71)(61,78)(63,67)(65,66)(76,77)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=g,e^2=hi,f^2=gh,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, 41842);