Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93814 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7080 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| prime radical | C23 | #359 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 128 points:
(1,2,5,12,8,14,33,59,9,16,34,61,38,15,65,60)(3,17,19,67,22,68,75,121,23,70,76,103,42,69,95,122)(4,24,27,80,30,81,87,126,31,66,88,115,90,82,111,127)(6,35,37,97,39,64,100,57,7,40,32,93,26,13,62,58)(10,43,44,72,46,101,106,73,48,102,55,84,47,110,107,18)(11,49,51,86,52,112,116,28,54,89,56,91,53,118,117,29)(20,77,78,125,36,98,99,114,21,79,74,124,41,50,94,113)(25,63,83,119,85,45,108,105,71,109,123,120,92,96,128,104);(1,3,9,23)(2,10,16,48)(4,25,31,71)(5,19,34,76)(6,36,7,41)(8,42,38,22)(11,50,54,98)(12,55,61,44)(13,63,64,109)(14,47,15,46)(17,66,70,24)(18,28,73,29)(20,26,21,39)(27,83,88,123)(30,92,90,85)(32,94,37,99)(33,95,65,75)(35,96,40,45)(43,89,102,49)(51,114,56,113)(52,79,53,77)(57,119,58,120)(59,106,60,107)(62,74,100,78)(67,80,103,115)(68,81,69,82)(72,86,84,91)(87,128,111,108)(93,104,97,105)(101,112,110,118)(116,125,117,124)(121,127,122,126)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=g,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1011);