Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10512 |
| commutator, Frattini, cosocle | #405 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
214 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,8,13)(3,15,20,60)(4,21,26,75)(5,27,32,57)(6,12,34,55)(7,36,29,14)(9,38,41,17)(10,43,48,24)(11,49,52,31)(16,61,65,120)(18,59,68,117)(19,70,37,50)(22,76,79,119)(23,80,84,112)(25,53,62,74)(28,56,89,42)(30,72,92,115)(33,47,86,46)(35,94,90,54)(39,97,100,121)(40,96,102,66)(44,105,99,126)(45,91,108,127)(51,103,113,85)(58,93,116,64)(63,98,81,125)(67,110,78,106)(69,122,95,111)(71,123,124,77)(73,109,118,83)(82,87,104,107)(88,114,128,101);(1,3,8,20)(2,9,13,41)(4,22,26,79)(5,28,32,89)(6,18,34,68)(7,37,29,19)(10,44,48,99)(11,50,52,70)(12,40,55,102)(14,56,36,42)(15,57,60,27)(16,62,65,25)(17,49,38,31)(21,71,75,124)(23,81,84,63)(24,78,43,67)(30,88,92,128)(33,93,86,64)(35,95,90,69)(39,47,100,46)(45,58,108,116)(51,111,113,122)(53,98,74,125)(54,114,94,101)(59,115,117,72)(61,118,120,73)(66,103,96,85)(76,112,119,80)(77,109,123,83)(82,97,104,121)(87,106,107,110)(91,105,127,126);(1,4,5,23,29,62,49,109,8,26,32,84,7,25,31,83)(2,10,11,45,36,46,57,107,13,48,52,108,14,47,27,87)(3,16,17,63,37,79,89,123,20,65,38,81,19,22,28,77)(6,33,30,91,90,43,103,104,34,86,92,127,35,24,85,82)(9,39,15,58,56,44,70,106,41,100,60,116,42,99,50,110)(12,53,51,112,94,75,115,118,55,74,113,80,54,21,72,73)(18,67,66,105,95,93,128,97,68,78,96,126,69,64,88,121)(40,71,59,76,114,125,122,120,102,124,117,119,101,98,111,61)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,e^c=egh,f^c=fgh,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52694);