Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2233 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#41 | D8 |
| 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 | |
352 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,9,16,39)(4,20,21,74)(5,25,31,87)(6,32,35,97)(7,36,34,13)(10,43,44,22)(11,47,51,29)(12,52,54,28)(15,56,57,72)(17,63,26,84)(18,68,61,110)(19,70,62,42)(23,82,77,112)(24,83,37,76)(27,92,80,53)(30,73,95,86)(33,75,98,49)(38,78,100,58)(40,90,48,66)(41,89,103,65)(45,93,108,96)(46,81,55,99)(50,94,85,79)(59,118,115,127)(60,119,71,114)(64,122,88,106)(67,113,91,121)(69,109,120,104)(101,123,128,125)(102,117,107,126)(105,124,111,116);(1,3)(2,9)(4,15)(5,26)(6,18)(7,19)(8,16)(10,38)(11,48)(12,41)(13,42)(14,39)(17,31)(20,72)(21,57)(22,78)(23,59)(24,60)(25,84)(27,88)(28,65)(29,90)(30,91)(32,68)(33,69)(34,62)(35,61)(36,70)(37,71)(40,51)(43,58)(44,100)(45,101)(46,102)(47,66)(49,109)(50,111)(52,89)(53,106)(54,103)(55,107)(56,74)(63,87)(64,80)(67,95)(73,121)(75,104)(76,119)(77,115)(79,124)(81,126)(82,127)(83,114)(85,105)(86,113)(92,122)(93,125)(94,116)(96,123)(97,110)(98,120)(99,117)(108,128)(112,118);(1,4,7,37,8,21,34,24)(2,10,13,55,14,44,36,46)(3,15,19,71,16,57,62,60)(5,27,30,96,31,80,95,93)(6,33,28,23,35,98,52,77)(9,38,42,107,39,100,70,102)(11,49,50,82,51,75,85,112)(12,53,32,45,54,92,97,108)(17,64,67,125,26,88,91,123)(18,69,65,59,61,120,89,115)(20,73,83,25,74,86,76,87)(22,79,99,29,43,94,81,47)(40,104,105,118,48,109,111,127)(41,106,68,101,103,122,110,128)(56,113,119,63,72,121,114,84)(58,116,126,66,78,124,117,90);(3,16)(9,39)(15,57)(17,26)(18,61)(19,62)(38,100)(40,48)(41,103)(42,70)(56,72)(58,78)(59,115)(60,71)(63,84)(64,88)(65,89)(66,90)(67,91)(68,110)(69,120)(101,128)(102,107)(104,109)(105,111)(106,122)(113,121)(114,119)(116,124)(117,126)(118,127)(123,125)Polycyclic presentation:
2^9|a^2=i,c^2=eh,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fgi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420188);