Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150878 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11886 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,6,14,10,16,9,47)(3,17,20,69,24,70,23,83)(4,25,29,85,33,86,32,99)(5,34,38,101,41,67,42,102)(7,43,27,66,8,45,37,15)(11,48,50,79,53,80,52,19)(12,54,57,30,58,91,109,31)(13,59,62,39,64,94,63,40)(18,71,73,122,76,116,75,124)(21,81,44,108,22,82,46,60)(26,87,89,120,74,68,90,121)(28,92,95,56,36,104,98,61)(35,65,103,107,78,111,106,51)(49,110,112,123,114,128,113,88)(55,93,117,72,119,105,118,77)(84,125,126,96,100,115,127,97);(1,3,10,24)(2,11,16,53)(4,26,33,74)(5,35,41,78)(6,23,9,20)(7,44,8,46)(12,55,58,119)(13,60,64,108)(14,50,47,52)(15,65,66,111)(17,67,70,34)(18,31,76,30)(19,39,79,40)(21,37,22,27)(25,49,86,114)(28,93,36,105)(29,90,32,89)(38,106,42,103)(43,107,45,51)(48,59,80,94)(54,73,91,75)(56,120,61,121)(57,117,109,118)(62,81,63,82)(68,92,87,104)(69,102,83,101)(71,84,116,100)(72,95,77,98)(85,112,99,113)(88,96,123,97)(110,125,128,115)(122,126,124,127);(1,4,27,91,6,29,8,31,10,33,37,54,9,32,7,30)(2,12,43,85,47,109,15,25,16,58,45,99,14,57,66,86)(3,18,44,89,20,73,22,74,24,76,46,90,23,75,21,26)(5,36,59,115,42,95,40,96,41,28,94,125,38,98,39,97)(11,49,111,118,50,112,51,55,53,114,65,117,52,113,107,119)(13,61,102,126,62,92,34,100,64,56,101,127,63,104,67,84)(17,68,81,122,83,120,60,71,70,87,82,124,69,121,108,116)(19,77,103,110,80,105,35,88,79,72,106,128,48,93,78,123)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=gh,d^2=ghi,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58078);