Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
735 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 735 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 128 points:
(1,2,8,14)(3,15,21,59)(4,22,27,45)(5,28,31,11)(6,32,35,92)(7,36,38,85)(9,39,44,18)(10,23,48,25)(12,50,52,34)(13,53,55,30)(16,60,65,113)(17,58,20,57)(19,69,70,109)(24,76,80,123)(26,47,83,108)(29,86,89,91)(33,54,94,101)(37,51,67,112)(40,97,100,118)(41,66,43,68)(42,87,102,93)(46,104,106,126)(49,88,84,111)(56,61,114,63)(62,71,96,98)(64,119,120,105)(72,110,103,79)(73,81,95,75)(74,121,78,90)(77,115,125,99)(82,116,117,128)(107,122,127,124);(1,3)(2,9)(4,16)(5,17)(6,33)(7,37)(8,21)(10,40)(11,41)(12,51)(13,54)(14,44)(15,32)(18,34)(19,30)(20,31)(22,71)(23,61)(24,77)(25,63)(26,82)(27,65)(28,43)(29,87)(35,94)(36,42)(38,67)(39,50)(45,98)(46,105)(47,107)(48,100)(49,109)(52,112)(53,70)(55,101)(56,90)(57,91)(58,86)(59,92)(60,76)(62,79)(64,75)(66,88)(68,111)(69,84)(72,116)(73,115)(74,122)(78,124)(80,125)(81,120)(83,117)(85,102)(89,93)(95,99)(96,110)(97,104)(103,128)(106,119)(108,127)(113,123)(114,121)(118,126);(1,4)(2,10)(3,16)(5,23)(6,24)(7,26)(8,27)(9,40)(11,45)(12,46)(13,47)(14,48)(15,56)(17,61)(18,62)(19,64)(20,63)(21,65)(22,28)(25,31)(29,74)(30,75)(32,90)(33,77)(34,79)(35,80)(36,95)(37,82)(38,83)(39,96)(41,98)(42,99)(43,71)(44,100)(49,103)(50,110)(51,105)(52,106)(53,81)(54,107)(55,108)(57,113)(58,60)(59,114)(66,97)(67,117)(68,118)(69,116)(70,120)(72,84)(73,85)(76,86)(78,89)(87,122)(88,104)(91,123)(92,121)(93,124)(94,125)(101,127)(102,115)(109,128)(111,126)(112,119);(1,5,8,31)(2,11,14,28)(3,17,21,20)(4,23,27,25)(6,29,35,89)(7,30,38,53)(9,41,44,43)(10,45,48,22)(12,49,52,84)(13,36,55,85)(15,57,59,58)(16,61,65,63)(18,66,39,68)(19,67,70,37)(24,74,80,78)(26,75,83,81)(32,91,92,86)(33,87,94,93)(34,88,50,111)(40,98,100,71)(42,101,102,54)(46,103,106,72)(47,95,108,73)(51,109,112,69)(56,113,114,60)(62,97,96,118)(64,117,120,82)(76,90,123,121)(77,122,125,124)(79,104,110,126)(99,127,115,107)(105,128,119,116)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420048);