Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #508 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 132 points:
(2,4)(5,6,11,19,14,21,13,49)(7,22,27,68,29,61,46,52)(8,30,34,57,37,90,36,101)(9,38,41,50,43,103,10,44)(12,47,40,63,48,62,45,20)(15,26,53,25,56,82,55,39)(16,32,58,94,60,96,106,33)(17,23,64,72,65,31,18,42)(24,73,77,116,79,123,98,122)(28,85,84,69,81,107,80,102)(35,99,93,108,100,129,97,89)(51,109,110,126,112,125,111,78)(54,83,67,70,113,104,105,74)(59,91,88,95,87,71,115,120)(66,75,117,127,114,92,86,76)(118,119,132,130,131,124,121,128);(1,2,3,4)(5,7,23,70,12,28,43,25,11,46,72,83,40,80,10,26,14,29,31,74,48,81,9,39,13,27,42,104,45,84,41,82)(6,15,50,107,20,54,17,61,49,53,38,102,62,67,18,52,21,56,44,85,63,113,65,22,19,55,103,69,47,105,64,68)(8,24,71,119,35,78,96,75,34,98,120,128,93,125,33,76,37,79,91,124,100,126,32,92,36,77,95,130,97,109,94,127)(16,51,108,132,59,73,57,114,106,110,99,131,115,116,30,86,60,112,89,121,87,123,101,66,58,111,129,118,88,122,90,117);(5,8,31,91,12,35,9,32,11,34,42,95,40,93,41,94,14,37,23,71,48,100,43,96,13,36,72,120,45,97,10,33)(6,16,44,89,20,59,65,101,49,106,103,129,62,115,64,90,21,60,50,108,63,87,17,57,19,58,38,99,47,88,18,30)(7,24,74,124,28,78,39,92,46,98,104,130,80,125,82,127,29,79,70,119,81,126,25,75,27,77,83,128,84,109,26,76)(15,51,85,121,54,73,22,66,53,110,69,118,67,116,68,117,56,112,107,132,113,123,61,114,55,111,102,131,105,122,52,86)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,d^b=di,f^b=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60174);