Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509294 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30567 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,16,9,13,6,14)(3,17,22,44,23,10,20,46)(4,24,28,50,27,61,41,11)(5,30,31,12,33,53,34,52)(7,15,39,60,40,55,35,56)(18,64,68,42,67,104,78,62)(19,69,70,51,72,92,73,43)(21,63,36,98,77,45,74,57)(25,79,84,47,83,108,75,109)(26,48,89,80,86,112,87,110)(29,81,90,102,65,58,37,49)(32,93,94,54,38,101,97,59)(66,114,122,119,120,103,121,127)(71,118,95,106,76,115,99,105)(82,126,123,125,85,107,124,128)(88,111,100,117,91,113,96,116);(1,3,9,23)(2,10,13,17)(4,25,27,83)(5,19,33,72)(6,22,8,20)(7,36,40,74)(11,47,50,109)(12,51,52,43)(14,46,16,44)(15,57,55,98)(18,65,67,29)(21,35,77,39)(24,79,61,108)(26,85,86,82)(28,75,41,84)(30,92,53,69)(31,73,34,70)(32,95,38,99)(37,68,90,78)(42,102,62,49)(45,56,63,60)(48,107,112,126)(54,115,59,118)(58,104,81,64)(66,91,120,88)(71,97,76,94)(80,125,110,128)(87,124,89,123)(93,105,101,106)(96,122,100,121)(103,111,114,113)(116,119,117,127);(1,4,8,41,9,27,6,28)(2,11,14,24,13,50,16,61)(3,18,22,78,23,67,20,68)(5,26,31,87,33,86,34,89)(7,37,39,65,40,90,35,29)(10,42,44,104,17,62,46,64)(12,48,30,80,52,112,53,110)(15,58,56,49,55,81,60,102)(19,66,70,121,72,120,73,122)(21,75,36,83,77,84,74,25)(32,96,94,91,38,100,97,88)(43,103,92,127,51,114,69,119)(45,79,98,47,63,108,57,109)(54,116,93,111,59,117,101,113)(71,123,95,82,76,124,99,85)(105,128,115,126,106,125,118,107);(1,5,9,33)(2,12,13,52)(3,19,23,72)(4,26,27,86)(6,34,8,31)(7,38,40,32)(10,43,17,51)(11,48,50,112)(14,30,16,53)(15,59,55,54)(18,66,67,120)(20,73,22,70)(21,76,77,71)(24,80,61,110)(25,82,83,85)(28,89,41,87)(29,91,65,88)(35,94,39,97)(36,99,74,95)(37,100,90,96)(42,103,62,114)(44,92,46,69)(45,106,63,105)(47,107,109,126)(49,113,102,111)(56,101,60,93)(57,115,98,118)(58,117,81,116)(64,119,104,127)(68,122,78,121)(75,124,84,123)(79,125,108,128)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=ghi,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416493);