Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13407 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,19)(4,27)(5,17)(6,36)(7,42)(8,46)(9,13)(10,18)(11,21)(12,30)(14,40)(15,45)(16,39)(20,57)(22,68)(23,64)(24,54)(25,56)(26,55)(28,94)(29,98)(31,58)(32,69)(33,93)(34,106)(35,66)(37,74)(38,108)(41,107)(43,111)(44,91)(47,75)(48,92)(49,63)(50,59)(51,97)(52,87)(53,95)(60,110)(61,102)(62,100)(65,103)(67,83)(70,82)(71,101)(72,76)(73,78)(77,112)(79,114)(80,123)(81,117)(84,113)(85,125)(86,122)(88,116)(89,115)(90,120)(96,126)(99,119)(104,121)(105,124)(109,128)(118,127);(1,3,10,26)(2,11,18,56)(4,28,33,97)(5,25,9,21)(6,37,34,95)(7,43,41,87)(8,47,49,82)(12,57,61,123)(13,55,17,19)(14,64,67,125)(15,68,62,117)(16,70,66,75)(20,76,80,31)(22,83,81,40)(23,45,85,100)(24,35,89,39)(27,73,93,119)(29,99,50,78)(30,90,102,77)(32,104,105,84)(36,52,106,111)(38,109,101,79)(42,74,107,53)(44,88,110,96)(46,54,63,115)(48,86,103,127)(51,98,94,59)(58,112,72,120)(60,114,91,128)(65,113,92,121)(69,118,124,122)(71,116,108,126);(1,4,5,29,10,33,9,50)(2,12,13,58,18,61,17,72)(3,20,21,77,26,80,25,90)(6,38,7,44,34,101,41,110)(8,48,35,105,49,103,39,32)(11,51,19,73,56,94,55,119)(14,65,62,124,67,92,15,69)(16,71,63,91,66,108,46,60)(22,84,23,86,81,104,85,127)(24,88,82,109,89,96,47,79)(27,42,59,36,93,107,98,106)(28,95,78,43,97,37,99,87)(30,100,76,40,102,45,31,83)(52,114,74,126,111,128,53,116)(54,118,75,121,115,122,70,113)(57,64,120,68,123,125,112,117)Polycyclic presentation:
2^9|b^2=i,c^2=d,d^2=i,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fgi,g^b=gh,h^b=hi,e^c=egi,f^c=fg,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, 60471);