Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9762 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,7,14,8,15,6,39,10,16,41,65,9,46,42,55)(3,17,21,68,22,69,84,67,23,70,40,107,45,60,43,108)(4,24,27,87,28,88,95,12,30,57,44,58,29,98,26,59)(5,31,35,100,36,63,74,48,38,101,18,71,37,13,34,62)(11,47,51,83,52,72,64,104,54,103,112,20,53,82,50,32)(19,75,78,66,79,123,109,120,80,126,77,113,97,124,96,122)(25,89,92,61,93,111,91,99,33,105,73,56,94,86,106,121)(49,81,115,125,116,102,114,128,118,76,119,127,117,90,85,110);(1,3,18,72,8,45,34,47,10,23,35,82,9,22,74,103)(2,11,48,107,46,52,100,108,16,54,62,68,15,53,71,67)(4,19,73,125,28,97,106,110,30,80,92,127,29,79,91,128)(5,32,7,43,36,20,6,40,38,104,41,84,37,83,42,21)(12,49,111,123,87,116,89,126,59,118,86,124,58,117,105,75)(13,60,65,50,101,17,39,51,63,69,14,64,31,70,55,112)(24,85,99,113,98,115,61,122,57,114,121,66,88,119,56,120)(25,90,27,96,93,76,95,77,33,102,44,109,94,81,26,78);(1,4,7,44,8,28,6,26,10,30,41,27,9,29,42,95)(2,12,55,98,46,87,65,57,16,59,39,88,15,58,14,24)(3,19,43,109,45,97,40,78,23,80,84,96,22,79,21,77)(5,33,18,73,36,94,34,106,38,25,35,92,37,93,74,91)(11,49,112,115,52,116,50,114,54,118,51,119,53,117,64,85)(13,61,100,89,101,121,62,86,63,56,71,105,31,99,48,111)(17,66,68,124,69,120,67,75,70,113,107,123,60,122,108,126)(20,81,47,110,104,90,82,127,83,76,103,128,32,102,72,125)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=egi,d^2=ghi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48571);