Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152543 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13012 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,6,14,40,61,99,63,8,16,37,12,45,62,86,76,9,17,42,64,35,60,36,67,19,73,5,13,7,15,41,70)(3,18,23,50,43,68,33,10,25,49,21,71,89,65,107,51,26,79,90,117,88,123,92,52,46,109,22,78,24,58,38,74)(4,27,29,97,44,69,102,57,31,98,28,96,101,66,39,75,32,53,103,55,100,121,34,11,47,54,104,72,30,59,106,56)(20,48,82,111,94,116,87,119,84,113,81,110,127,125,93,115,85,114,108,124,126,128,91,77,95,118,80,122,83,112,105,120);(1,3,19,25,9,26,8,46)(2,10,16,74,17,52,73,51)(4,20,31,95,32,85,47,84)(5,33,42,107,37,92,6,38)(7,43,35,89,45,88,40,24)(11,48,75,113,57,114,56,118)(12,58,64,123,13,65,14,68)(15,71,61,50,62,78,60,117)(18,76,109,67,79,70,49,63)(21,86,90,99,22,41,23,36)(27,77,54,120,53,119,98,115)(28,91,103,93,104,87,29,105)(30,94,44,83,101,126,100,127)(34,81,106,82,102,80,39,108)(55,116,96,125,97,128,72,112)(59,122,121,111,66,110,69,124);(1,4)(2,11)(3,20)(5,34)(6,39)(7,44)(8,47)(9,32)(10,48)(12,59)(13,66)(14,69)(15,72)(16,75)(17,57)(18,77)(19,31)(21,87)(22,91)(23,93)(24,94)(25,95)(26,85)(27,63)(28,99)(29,86)(30,40)(33,81)(35,101)(36,104)(37,102)(38,108)(41,103)(42,106)(43,83)(45,100)(46,84)(49,115)(50,116)(51,118)(52,114)(53,67)(54,76)(55,61)(56,73)(58,122)(60,97)(62,96)(64,121)(65,110)(68,124)(70,98)(71,112)(74,113)(78,125)(79,119)(80,92)(82,107)(88,127)(89,126)(90,105)(109,120)(111,123)(117,128)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,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, 59739);