Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512698 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#357 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 272 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 — 4 generators on 128 points:
(1,2,6,13,9,15,8,39)(3,16,19,59,22,61,21,73)(4,11,25,47,23,45,27,85)(5,28,31,86,34,88,33,93)(7,35,30,56,38,51,37,14)(10,40,42,18,44,68,54,70)(12,49,52,92,53,91,87,32)(17,57,64,110,62,108,66,118)(20,71,67,50,72,107,36,60)(24,76,79,121,74,119,81,126)(26,82,78,100,75,103,84,48)(29,55,89,98,69,58,90,43)(41,94,96,63,95,115,105,112)(46,101,104,120,99,125,122,80)(65,117,114,102,113,128,83,111)(77,106,123,127,116,109,124,97);(1,3,9,22)(2,10,15,44)(4,17,23,62)(5,29,34,69)(6,21,8,19)(7,36,38,67)(11,41,45,95)(12,50,53,60)(13,54,39,42)(14,55,56,58)(16,28,61,88)(18,32,70,92)(20,30,72,37)(24,77,74,116)(25,66,27,64)(26,83,75,114)(31,90,33,89)(35,43,51,98)(40,49,68,91)(46,102,99,111)(47,105,85,96)(48,106,100,109)(52,71,87,107)(57,76,108,119)(59,93,73,86)(63,80,112,120)(65,78,113,84)(79,124,81,123)(82,97,103,127)(94,101,115,125)(104,117,122,128)(110,126,118,121);(1,4)(2,11)(3,17)(5,24)(6,25)(7,26)(8,27)(9,23)(10,41)(12,46)(13,47)(14,48)(15,45)(16,57)(18,63)(19,64)(20,65)(21,66)(22,62)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,74)(35,82)(36,83)(37,84)(38,75)(39,85)(40,94)(42,96)(43,97)(44,95)(49,101)(50,102)(51,103)(52,104)(53,99)(54,105)(55,106)(56,100)(58,109)(59,110)(60,111)(61,108)(67,114)(68,115)(69,116)(70,112)(71,117)(72,113)(73,118)(86,121)(87,122)(88,119)(89,123)(90,124)(91,125)(92,120)(93,126)(98,127)(107,128);(4,23)(11,45)(17,62)(24,74)(25,27)(26,75)(41,95)(46,99)(47,85)(48,100)(57,108)(63,112)(64,66)(65,113)(76,119)(77,116)(78,84)(79,81)(80,120)(82,103)(83,114)(94,115)(96,105)(97,127)(101,125)(102,111)(104,122)(106,109)(110,118)(117,128)(121,126)(123,124)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419899);