Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152540 |
| 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:
(1,2,6,13,40,66,102,62,8,15,37,12,45,61,83,73,9,16,42,63,35,60,36,93,18,70,5,32,7,14,41,67)(3,17,21,50,43,64,33,10,23,49,20,68,86,104,105,51,24,76,87,118,85,120,106,52,46,109,88,75,22,58,38,71)(4,25,28,95,44,108,34,57,30,97,26,94,99,59,107,72,31,53,100,55,98,65,103,11,47,54,27,69,29,96,39,56)(19,48,79,112,90,117,101,74,81,114,78,110,124,122,126,121,82,115,125,127,123,128,84,92,91,119,77,111,80,113,89,116);(1,3,18,23,9,24,8,46)(2,10,15,71,16,52,70,51)(4,19,30,91,31,82,47,81)(5,33,42,105,37,106,6,38)(7,43,35,86,45,85,40,22)(11,48,72,114,57,115,56,119)(12,58,63,120,32,104,13,64)(14,68,66,50,61,75,60,118)(17,73,109,93,76,67,49,62)(20,83,87,102,88,41,21,36)(25,92,54,116,53,74,97,121)(26,84,100,126,27,101,28,89)(29,90,44,80,99,123,98,124)(34,77,107,125,103,78,39,79)(55,117,94,122,95,128,69,113)(59,110,108,127,96,111,65,112);(1,4)(2,11)(3,19)(5,34)(6,39)(7,44)(8,47)(9,31)(10,48)(12,59)(13,65)(14,69)(15,72)(16,57)(17,74)(18,30)(20,84)(21,89)(22,90)(23,91)(24,82)(25,93)(26,36)(27,102)(28,41)(29,40)(32,96)(33,77)(35,99)(37,103)(38,79)(42,107)(43,80)(45,98)(46,81)(49,116)(50,117)(51,119)(52,115)(53,62)(54,67)(55,66)(56,70)(58,110)(60,95)(61,94)(63,108)(64,112)(68,113)(71,114)(73,97)(75,122)(76,92)(78,106)(83,100)(85,124)(86,123)(87,126)(88,101)(104,111)(105,125)(109,121)(118,128)(120,127)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,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gh,h^b=hi,d^c=dg,e^c=egi,f^c=fg,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, 59736);