Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10564 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,2,9,16)(3,17,23,71)(4,24,30,65)(5,31,37,67)(6,38,7,41)(8,45,33,15)(10,47,52,19)(11,43,57,27)(12,58,62,36)(13,40,14,44)(18,72,76,124)(20,79,21,80)(22,82,46,59)(25,68,81,123)(26,89,94,121)(28,85,29,84)(32,66,103,51)(34,60,35,61)(39,70,42,69)(48,108,111,86)(49,101,50,102)(53,107,112,73)(54,115,118,128)(55,95,56,96)(63,77,64,78)(74,125,75,116)(83,93,120,92)(87,122,88,119)(90,110,127,109)(91,97,105,117)(98,104,99,106)(100,114,126,113);(1,3,9,23)(2,10,16,52)(4,25,30,81)(5,32,37,103)(6,39,7,42)(8,46,33,22)(11,53,57,112)(12,59,62,82)(13,63,14,64)(15,66,45,51)(17,67,71,31)(18,43,76,27)(19,58,47,36)(20,44,21,40)(24,48,65,111)(26,90,94,127)(28,88,29,87)(34,78,35,77)(38,50,41,49)(54,116,118,125)(55,114,56,113)(60,70,61,69)(68,121,123,89)(72,117,124,97)(73,115,107,128)(74,95,75,96)(79,98,80,99)(83,119,120,122)(84,110,85,109)(86,93,108,92)(91,100,105,126)(101,106,102,104);(1,4,5,26,33,29,58,93,9,30,37,94,8,28,36,92)(2,11,12,54,45,56,67,117,16,57,62,118,15,55,31,97)(3,18,19,73,46,75,103,126,23,76,47,107,22,74,32,100)(6,27,34,91,44,96,106,115,7,43,35,105,40,95,104,128)(10,48,17,68,66,110,82,122,52,111,71,123,51,109,59,119)(13,24,60,83,38,85,99,121,14,65,61,120,41,84,98,89)(20,25,77,86,42,88,102,127,21,81,78,108,39,87,101,90)(49,53,69,72,64,114,79,125,50,112,70,124,63,113,80,116)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52943);