Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #512493 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33272 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | #578 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,16)(3,17,20,10)(4,23,29,74)(5,30,34,55)(6,35,37,52)(7,15,8,14)(11,44,48,26)(12,39,51,33)(13,36,53,24)(18,59,62,70)(19,49,65,96)(21,69,68,90)(22,43,38,54)(25,77,81,121)(27,73,28,72)(31,41,67,57)(32,88,89,50)(40,63,56,97)(42,60,58,91)(45,99,101,122)(46,82,47,75)(61,111,112,128)(64,115,83,109)(66,107,116,95)(71,84,120,80)(76,86,92,105)(78,119,114,108)(79,85,98,103)(87,104,117,118)(93,125,106,123)(94,124,102,110)(100,127,126,113);(1,3,7,38,9,20,8,22)(2,10,14,54,16,17,15,43)(4,18,27,83,29,62,28,64)(5,31,33,65,34,67,39,19)(6,21,36,91,37,68,24,60)(11,40,46,102,48,56,47,94)(12,49,30,57,51,96,55,41)(13,42,52,69,53,58,35,90)(23,70,72,109,74,59,73,115)(25,78,80,112,81,114,84,61)(26,63,82,124,44,97,75,110)(32,87,76,116,89,117,92,66)(45,100,85,106,101,126,103,93)(50,104,86,107,88,118,105,95)(71,111,77,108,120,128,121,119)(79,123,122,127,98,125,99,113);(1,4,24,75,8,28,37,44,9,29,36,82,7,27,6,26)(2,11,35,73,15,47,53,74,16,48,52,72,14,46,13,23)(3,18,60,110,22,64,68,97,20,62,91,124,38,83,21,63)(5,25,76,122,33,80,89,98,34,81,92,99,39,84,32,79)(10,40,90,115,43,94,58,59,17,56,69,109,54,102,42,70)(12,45,86,77,30,85,88,120,51,101,105,121,55,103,50,71)(19,61,87,123,31,78,116,127,65,112,117,125,67,114,66,113)(41,93,104,111,49,100,107,108,57,106,118,128,96,126,95,119);(1,5)(2,12)(3,19,20,65)(4,25)(6,32)(7,39)(8,33)(9,34)(10,41,17,57)(11,45)(13,50)(14,55)(15,30)(16,51)(18,61,62,112)(21,66,68,116)(22,31,38,67)(23,71)(24,76)(26,79)(27,84)(28,80)(29,81)(35,86)(36,92)(37,89)(40,93,56,106)(42,95,58,107)(43,49,54,96)(44,98)(46,103)(47,85)(48,101)(52,105)(53,88)(59,108,70,119)(60,87,91,117)(63,113,97,127)(64,78,83,114)(69,118,90,104)(72,121)(73,77)(74,120)(75,122)(82,99)(94,100,102,126)(109,128,115,111)(110,123,124,125)Polycyclic presentation:
2^9|a^2=i,b^2=eg,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,g^b=gi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419690);