Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152543 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13012 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,12,32,56,96,113,8,15,39,59,7,14,38,58,9,16,5,13,33,57,95,114,42,66,35,63,43,67,34,62)(3,17,20,71,80,115,93,48,22,72,83,47,21,54,36,49,23,73,19,64,40,99,97,68,44,101,81,108,85,60,30,10)(4,24,26,65,89,116,98,69,28,88,91,111,27,55,31,11,29,50,25,87,41,100,94,52,45,102,90,51,92,61,37,53)(18,46,75,103,121,127,126,118,77,105,123,117,76,104,79,110,78,106,74,112,84,107,125,120,86,109,122,128,124,119,82,70);(1,3)(2,10,16,49)(4,18)(5,30)(6,36)(7,40)(8,44)(9,23)(11,46,53,106)(12,54,13,60)(14,64,67,71)(15,68,66,48)(17,62,73,58)(19,34)(20,38)(21,33)(22,42)(24,70,50,110)(25,82)(26,79)(27,84)(28,86)(29,78)(31,75)(32,85)(35,97)(37,74)(39,93)(41,76)(43,80)(45,77)(47,56,108,57)(51,107,111,127)(52,109,69,105)(55,112,61,103)(59,115,63,99)(65,104,87,119)(72,114,101,113)(81,95)(83,96)(88,120,102,118)(89,124)(90,125)(91,126)(92,121)(94,122)(98,123)(100,117,116,128);(1,4,9,29)(2,11,16,53)(3,18,23,78)(5,31,6,37)(7,41,43,89)(8,45,42,28)(10,46,49,106)(12,55,13,61)(14,65,67,87)(15,69,66,52)(17,70,73,110)(19,79,20,82)(21,84,85,121)(22,86,44,77)(24,62,50,58)(25,34,26,38)(27,32,92,33)(30,75,36,74)(35,98,39,94)(40,76,80,124)(47,107,108,127)(48,109,68,105)(51,56,111,57)(54,112,60,103)(59,116,63,100)(64,104,71,119)(72,120,101,118)(81,126,83,125)(88,114,102,113)(90,95,91,96)(93,122,97,123)(99,117,115,128)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=ghi,e^2=ghi,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59742);