Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508805 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30137 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
412 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)(3,10)(4,21)(5,28,30,53)(6,33)(7,14)(8,13)(9,15)(11,24)(12,31,52,34)(16,60)(17,65,67,84)(18,45)(19,44)(20,46)(22,74)(23,51,29,48)(25,58)(26,50)(27,55)(32,85,88,112)(35,92)(36,57)(37,94)(38,56)(39,59)(40,43)(41,63)(42,70,100,68)(47,72)(49,82)(54,89,111,93)(61,115)(62,96,66,99)(64,97)(69,113,121,126)(71,98)(73,104)(75,79,107,81)(76,102)(77,105,86,109)(78,103)(80,110,87,108)(83,106)(90,127,122,101)(91,116,95,114)(117,119,120,118)(123,125,124,128);(1,3,9,20)(2,10,15,46)(4,22,27,78)(5,17,30,67)(6,18,8,40)(7,36,39,76)(11,47,50,106)(12,42,52,100)(13,43,33,45)(14,57,59,102)(16,61,64,25)(19,35,73,38)(21,74,55,103)(23,77,29,86)(24,72,26,83)(28,84,53,65)(31,68,34,70)(32,90,88,122)(37,71,82,63)(41,94,98,49)(44,92,104,56)(48,105,51,109)(54,113,111,126)(58,60,115,97)(62,87,66,80)(69,93,121,89)(75,125,107,128)(79,123,81,124)(85,101,112,127)(91,119,95,118)(96,110,99,108)(114,120,116,117);(1,4,9,27)(2,11,15,50)(3,16,20,64)(5,29,30,23)(6,26,8,24)(7,37,39,82)(10,41,46,98)(12,51,52,48)(13,55,33,21)(14,58,59,115)(17,66,67,62)(18,71,40,63)(19,72,73,83)(22,76,78,36)(25,35,61,38)(28,75,53,107)(31,79,34,81)(32,91,88,95)(42,99,100,96)(43,97,45,60)(44,103,104,74)(47,102,106,57)(49,92,94,56)(54,114,111,116)(65,120,84,117)(68,118,70,119)(69,123,121,124)(77,90,86,122)(80,89,87,93)(85,110,112,108)(101,125,127,128)(105,113,109,126);(1,5,6,34,9,30,8,31)(2,12,13,28,15,52,33,53)(3,17,18,70,20,67,40,68)(4,23,24,81,27,29,26,79)(7,32,35,93,39,88,38,89)(10,42,43,84,46,100,45,65)(11,48,21,75,50,51,55,107)(14,54,56,85,59,111,92,112)(16,62,63,119,64,66,71,118)(19,69,36,90,73,121,76,122)(22,77,72,124,78,86,83,123)(25,80,82,91,61,87,37,95)(41,96,60,117,98,99,97,120)(44,101,102,126,104,127,57,113)(47,105,74,125,106,109,103,128)(49,108,58,116,94,110,115,114)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=ef,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416004);