Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153253 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13381 |
| 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 | |
260 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 260 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)(3,10,21,45)(4,22,24,77)(5,29,35,91)(6,15,8,13)(7,38,40,58)(11,46,48,25)(12,52,56,33)(14,37,60,31)(17,61,62,118)(18,65,70,124)(19,57,36,43)(20,72,74,115)(23,81,71,75)(26,80,28,78)(27,89,83,113)(30,92,86,59)(32,90,34,76)(39,79,88,53)(41,100,101,63)(42,103,106,69)(44,98,109,67)(47,82,107,110)(49,85,51,87)(50,95,111,94)(54,84,55,96)(64,122,99,112)(66,125,121,108)(68,117,97,123)(73,119,128,104)(93,116,127,102)(105,126,114,120);(1,3)(2,10)(4,23)(5,18)(6,36)(7,20)(8,19)(9,21)(11,47)(12,42)(13,57)(14,44)(15,43)(16,45)(17,26)(22,75)(24,71)(25,82)(27,73)(28,62)(29,65)(30,93)(31,67)(32,97)(33,69)(34,68)(35,70)(37,98)(38,72)(39,99)(40,74)(41,49)(46,110)(48,107)(50,108)(51,101)(52,103)(53,112)(54,114)(55,105)(56,106)(58,115)(59,116)(60,109)(61,78)(63,85)(64,88)(66,94)(76,117)(77,81)(79,122)(80,118)(83,128)(84,120)(86,127)(87,100)(89,104)(90,123)(91,124)(92,102)(95,121)(96,126)(111,125)(113,119);(1,4,8,26,9,24,6,28)(2,11,15,49,16,48,13,51)(3,17,19,71,21,62,36,23)(5,30,34,94,35,86,32,95)(7,39,31,27,40,88,37,83)(10,41,43,107,45,101,57,47)(12,53,55,89,56,79,54,113)(14,59,38,50,60,92,58,111)(18,66,68,127,70,121,97,93)(20,73,67,64,74,128,98,99)(22,76,78,29,77,90,80,91)(25,84,85,33,46,96,87,52)(42,104,105,122,106,119,114,112)(44,108,72,102,109,125,115,116)(61,117,81,65,118,123,75,124)(63,120,110,69,100,126,82,103)Polycyclic presentation:
2^9|a^2=i,c^2=dh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,f^b=fi,d^c=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, 60450);