Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93772 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7076 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2113 | C22 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2,5,12,9,17,37,65)(3,18,20,72,24,76,84,109)(4,25,28,86,32,87,94,119)(6,14,39,62,27,68,34,101)(7,13,35,67,41,53,102,61)(8,15,36,63,40,16,70,64)(10,44,45,95,50,108,113,80)(11,51,54,29,58,96,60,90)(19,49,78,59,81,48,107,112)(21,52,85,117,42,105,82,128)(22,73,83,118,38,103,104,123)(23,74,33,99,43,75,100,127)(26,66,88,125,89,47,115,111)(30,56,92,121,77,57,106,122)(31,55,93,120,97,71,116,124)(46,91,114,98,69,126,110,79);(1,3,9,24)(2,10,17,50)(4,26,32,89)(5,33,37,100)(6,38,27,22)(7,42,41,21)(8,43,40,23)(11,52,58,105)(12,59,65,112)(13,66,53,47)(14,69,68,46)(15,49,16,48)(18,71,76,55)(19,77,81,30)(20,70,84,36)(25,74,87,75)(28,91,94,126)(29,95,90,80)(31,98,97,79)(34,85,39,82)(35,83,102,104)(44,106,108,92)(45,63,113,64)(51,107,96,78)(54,118,60,123)(56,73,57,103)(61,114,67,110)(62,111,101,125)(72,119,109,86)(88,116,115,93)(99,120,127,124)(117,121,128,122)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fhi,d^2=i,e^2=hi,f^2=h,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=fhi,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, 971);