Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508036 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29594 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,15,18,9)(4,21,26,72)(5,11,28,47)(6,30,34,70)(7,35,32,13)(10,41,45,24)(12,49,51,33)(16,56,22,73)(17,53,61,38)(19,66,64,86)(20,40,65,55)(23,71,76,120)(25,78,36,50)(27,79,82,119)(29,84,81,48)(31,52,88,44)(37,58,42,74)(39,68,54,103)(43,96,97,75)(46,99,101,83)(57,107,60,110)(59,94,69,104)(62,115,113,122)(63,93,114,106)(67,90,87,95)(77,121,85,100)(80,102,123,98)(89,108,91,111)(92,117,105,128)(109,127,112,126)(116,125,118,124);(1,3,7,20,8,18,32,65)(2,9,13,40,14,15,35,55)(4,22,25,69,26,16,36,59)(5,17,29,63,28,61,81,114)(6,19,33,68,34,64,49,103)(10,42,44,95,45,37,52,90)(11,38,48,93,47,53,84,106)(12,39,30,86,51,54,70,66)(21,56,50,104,72,73,78,94)(23,60,77,112,76,57,85,109)(24,74,31,87,41,58,88,67)(27,62,83,117,82,113,99,128)(43,91,98,125,97,89,102,124)(46,92,79,122,101,105,119,115)(71,107,100,126,120,110,121,127)(75,111,80,116,96,108,123,118);(1,4,8,26)(2,10,14,45)(3,16,18,22)(5,23,28,76)(6,31,34,88)(7,36,32,25)(9,37,15,42)(11,43,47,97)(12,50,51,78)(13,52,35,44)(17,60,61,57)(19,67,64,87)(20,69,65,59)(21,70,72,30)(24,49,41,33)(27,80,82,123)(29,85,81,77)(38,91,53,89)(39,94,54,104)(40,95,55,90)(46,100,101,121)(48,102,84,98)(56,86,73,66)(58,103,74,68)(62,116,113,118)(63,109,114,112)(71,119,120,79)(75,99,96,83)(92,126,105,127)(93,124,106,125)(107,115,110,122)(108,117,111,128);(1,5)(2,11)(3,17)(4,23)(6,27)(7,29)(8,28)(9,38)(10,43)(12,46)(13,48)(14,47)(15,53)(16,57)(18,61)(19,62)(20,63)(21,71)(22,60)(24,75)(25,77)(26,76)(30,79)(31,80)(32,81)(33,83)(34,82)(35,84)(36,85)(37,89)(39,92)(40,93)(41,96)(42,91)(44,98)(45,97)(49,99)(50,100)(51,101)(52,102)(54,105)(55,106)(56,107)(58,108)(59,109)(64,113)(65,114)(66,115)(67,118)(68,117)(69,112)(70,119)(72,120)(73,110)(74,111)(78,121)(86,122)(87,116)(88,123)(90,124)(94,127)(95,125)(103,128)(104,126)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415235);