Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94613 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7385 |
| commutator | C32 | C2×C8 |
| Frattini, cosocle | #1397 | C22 |
| prime radical | C22 | #1419 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ #1397 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,7,14,33,59,9,16,34,60,41,15,8,42)(3,17,19,67,21,69,22,85,23,72,43,70,40,71,86,128)(4,24,27,87,29,65,30,105,31,91,32,89,104,90,106,127)(6,35,26,62,37,108,96,64,39,109,97,63,99,52,38,13)(10,44,45,18,47,75,117,77,49,115,118,78,48,79,58,111)(11,50,53,73,54,121,113,28,57,100,55,101,56,102,88,103)(20,81,80,51,82,122,83,68,84,125,74,124,112,123,36,110)(25,61,92,120,93,66,98,46,95,116,114,119,76,126,94,107);(1,3,9,23)(2,10,16,49)(4,25,31,95)(5,22,34,86)(6,36,39,83)(7,40,41,21)(8,43,33,19)(11,51,57,124)(12,58,60,117)(13,61,64,116)(14,48,15,47)(17,65,72,90)(18,73,78,101)(20,38,84,96)(24,71,91,69)(26,82,97,112)(27,98,32,94)(28,77,103,111)(29,76,104,93)(30,92,106,114)(35,107,109,46)(37,74,99,80)(42,45,59,118)(44,102,115,121)(50,75,100,79)(52,120,108,119)(53,81,55,125)(54,110,56,68)(62,126,63,66)(67,87,70,89)(85,127,128,105)(88,122,113,123)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=egh,d^2=f,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ceh,d^b=dh,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72321109, #72321119, #72321867, … |
| maximal normal subgroup of | #72321109, #72321119, #72321867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1811);