Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512779 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,16,20,62)(4,22)(5,12)(6,30)(7,36)(8,40)(9,15)(10,19,44,75)(11,23)(13,33)(14,38)(17,65,68,119)(18,60,41,64)(21,78,77,117)(24,84)(25,88)(26,47)(27,50)(28,92)(29,57)(31,56,76,61)(32,98)(34,101)(35,58)(37,63,74,52)(39,54)(42,104,107,124)(43,71,46,103)(45,102,109,93)(48,112)(49,90)(51,94)(53,118)(55,72)(59,66,106,70)(67,105,73,79)(69,116,125,113)(80,87)(81,83)(82,114)(85,108,97,121)(86,111)(89,122,123,126)(91,95)(96,100)(99,110)(115,128,120,127);(1,3)(2,10)(4,17)(5,18)(6,31)(7,37)(8,41)(9,20)(11,42)(12,43)(13,52)(14,56)(15,44)(16,58)(19,72)(21,38)(22,79)(23,66)(24,85)(25,89)(26,68)(27,70)(28,93)(29,77)(30,62)(32,97)(33,75)(34,102)(35,76)(36,45)(39,74)(40,46)(47,105)(48,113)(49,115)(50,107)(51,78)(53,116)(54,109)(55,63)(57,61)(59,114)(60,101)(64,92)(65,98)(67,99)(69,83)(71,100)(73,87)(80,126)(81,108)(82,127)(84,119)(86,125)(88,120)(90,123)(91,128)(94,103)(95,106)(96,117)(104,118)(110,122)(111,121)(112,124);(1,4,8,23,9,26,5,27)(2,11,12,22,15,50,40,47)(3,17,41,66,20,68,18,70)(6,32,34,91,35,24,28,82)(7,25,38,83,39,90,29,86)(10,42,43,79,44,107,46,105)(13,53,51,110,55,48,96,80)(14,49,36,81,57,88,54,111)(16,59,60,119,62,106,64,65)(19,73,103,124,75,67,71,104)(21,69,74,123,77,125,37,89)(30,95,92,98,58,114,101,84)(31,97,102,128,76,85,93,127)(33,99,100,118,72,87,94,112)(45,108,61,120,109,121,56,115)(52,116,78,122,63,113,117,126);(1,5,9,8)(2,12,15,40)(3,18,20,41)(4,23,26,27)(6,28,35,34)(7,29,39,38)(10,43,44,46)(11,47,50,22)(13,51,55,96)(14,36,57,54)(16,60,62,64)(17,66,68,70)(19,71,75,103)(21,37,77,74)(24,82,32,91)(25,83,90,86)(30,92,58,101)(31,93,76,102)(33,94,72,100)(42,105,107,79)(45,61,109,56)(48,110,53,80)(49,111,88,81)(52,78,63,117)(59,65,106,119)(67,104,73,124)(69,123,125,89)(84,114,98,95)(85,127,97,128)(87,112,99,118)(108,115,121,120)(113,122,116,126)Polycyclic presentation:
2^9|a^2=f,c^2=fh,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419976);