Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29732 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #585 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,7,13,33,52,80,73,8,14,36,55,6,12,32,51)(3,15,19,58,18,42,70,53,21,41,34,92,20,9,37,40)(4,22,25,50,31,90,91,45,26,75,81,44,24,54,35,10)(5,11,29,48,76,103,122,121,30,49,89,110,28,47,86,107)(16,59,62,38,61,97,93,96,64,111,79,98,63,72,71,56)(17,57,67,112,66,101,118,108,68,102,87,127,69,39,94,100)(23,74,77,106,85,125,126,104,78,109,88,46,27,82,84,43)(60,113,114,95,65,116,117,99,115,124,123,128,83,120,119,105);(1,3)(2,9)(4,16)(5,17)(6,20)(7,34)(8,21)(10,38)(11,39)(12,41)(13,53)(14,42)(15,52)(18,33)(19,36)(22,72)(23,60)(24,63)(25,79)(26,64)(27,83)(28,69)(29,87)(30,68)(31,61)(32,70)(35,93)(37,80)(40,55)(43,95)(44,56)(45,98)(46,105)(47,102)(48,108)(49,101)(50,96)(51,58)(54,111)(57,103)(59,90)(62,81)(65,85)(66,76)(67,89)(71,91)(73,92)(74,120)(75,97)(77,123)(78,115)(82,124)(84,117)(86,118)(88,114)(94,122)(99,106)(100,110)(104,128)(107,112)(109,116)(113,125)(119,126)(121,127);(1,4,8,26)(2,10,14,45)(3,16,21,64)(5,27,30,85)(6,31,33,24)(7,35,36,91)(9,38,42,98)(11,46,49,106)(12,50,52,44)(13,54,55,90)(15,56,41,96)(17,65,68,83)(18,63,20,61)(19,71,34,93)(22,73,75,51)(23,76,78,28)(25,80,81,32)(29,88,89,77)(37,62,70,79)(39,99,101,105)(40,59,53,111)(43,103,104,47)(48,109,110,74)(57,95,102,128)(58,72,92,97)(60,69,115,66)(67,114,87,123)(82,121,125,107)(84,122,126,86)(94,117,118,119)(100,116,108,120)(112,113,127,124);(1,5,6,28,8,30,33,76)(2,11,12,47,14,49,52,103)(3,17,20,69,21,68,18,66)(4,23,24,27,26,78,31,85)(7,29,32,86,36,89,80,122)(9,39,41,102,42,101,15,57)(10,43,44,46,45,104,50,106)(13,48,51,107,55,110,73,121)(16,60,63,83,64,115,61,65)(19,67,37,94,34,87,70,118)(22,74,54,82,75,109,90,125)(25,77,35,84,81,88,91,126)(38,95,56,105,98,128,96,99)(40,100,92,127,53,108,58,112)(59,113,72,120,111,124,97,116)(62,114,71,119,79,123,93,117)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415513);