Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153246 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13380 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 128 points:
(1,2,9,16)(3,17,22,51)(4,23,27,81)(5,15,8,12)(6,32,37,98)(7,38,40,53)(10,29,45,19)(11,46,49,26)(13,54,57,35)(14,31,59,34)(18,64,24,83)(20,71,74,127)(21,75,77,44)(25,86,41,80)(28,91,90,118)(30,79,36,97)(33,99,89,58)(39,82,103,55)(42,105,47,113)(43,109,111,125)(48,87,60,104)(50,94,115,101)(52,95,56,88)(61,84,108,66)(62,119,116,73)(63,93,117,70)(65,106,68,78)(67,120,85,110)(69,96,92,126)(72,114,100,107)(76,123,102,122)(112,124,121,128);(1,3)(2,10)(4,24)(5,29)(6,20)(7,21)(8,19)(9,22)(11,47)(12,51)(13,43)(14,44)(15,17)(16,45)(18,27)(23,78)(25,65)(26,84)(28,85)(30,92)(31,93)(32,96)(33,100)(34,70)(35,73)(36,69)(37,74)(38,63)(39,102)(40,77)(41,68)(42,49)(46,66)(48,61)(50,114)(52,116)(53,117)(54,119)(55,120)(56,62)(57,111)(58,121)(59,75)(60,108)(64,80)(67,90)(71,79)(72,89)(76,103)(81,106)(82,110)(83,86)(87,113)(88,125)(91,123)(94,124)(95,109)(97,127)(98,126)(99,112)(101,128)(104,105)(107,115)(118,122);(1,4,5,25,9,27,8,41)(2,11,12,48,16,49,15,60)(3,18,19,65,22,24,29,68)(6,33,30,94,37,89,36,101)(7,39,31,90,40,103,34,28)(10,42,17,61,45,47,51,108)(13,55,52,118,57,82,56,91)(14,58,53,115,59,99,38,50)(20,72,69,124,74,100,92,128)(21,76,70,67,77,102,93,85)(23,79,80,32,81,97,86,98)(26,88,87,35,46,95,104,54)(43,110,62,122,111,120,116,123)(44,112,63,107,75,121,117,114)(64,96,78,71,83,126,106,127)(66,109,113,73,84,125,105,119)Polycyclic presentation:
2^9|a^2=i,c^2=d,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60443);