Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499662 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23810 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 136 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:
(1,2,6,13,36,59,40,63,9,15,7,14,8,16,37,60)(3,17,20,64,41,61,38,45,22,68,76,46,21,57,34,10)(4,23,25,51,42,58,39,11,28,49,26,65,27,62,35,50)(5,12,30,53,91,113,94,116,32,56,31,54,33,55,92,114)(18,43,70,99,77,102,84,109,72,47,96,104,71,100,75,66)(19,67,73,110,88,118,97,101,74,111,89,52,29,85,87,44)(24,78,80,107,95,112,93,48,82,106,81,117,83,115,90,108)(69,98,120,127,86,105,126,128,121,103,125,123,79,124,122,119);(1,3)(2,10)(4,18)(5,29)(6,34)(7,38)(8,41)(9,22)(11,47)(12,52)(13,57)(14,61)(15,45)(16,64)(17,60)(19,33)(20,37)(21,36)(23,66)(24,79)(25,75)(26,84)(27,77)(28,72)(30,89)(31,73)(32,88)(35,70)(39,96)(40,76)(42,71)(43,50)(44,55)(46,59)(48,105)(49,109)(51,100)(53,111)(54,67)(56,110)(58,104)(62,99)(63,68)(65,102)(69,83)(74,91)(78,123)(80,125)(81,120)(82,86)(85,114)(87,92)(90,122)(93,126)(94,97)(95,121)(98,117)(101,113)(103,107)(106,127)(108,124)(112,128)(115,119)(116,118);(1,4,9,28)(2,11,15,50)(3,18,22,72)(5,24,32,82)(6,35,7,39)(8,42,36,27)(10,43,45,47)(12,48,56,108)(13,58,14,62)(16,65,59,51)(17,66,68,109)(19,69,74,121)(20,75,76,84)(21,77,41,71)(23,60,49,63)(25,40,26,37)(29,86,88,79)(30,90,31,93)(33,95,91,83)(34,70,38,96)(44,98,101,103)(46,102,64,100)(52,105,110,124)(53,112,54,115)(55,117,113,107)(57,99,61,104)(67,119,111,128)(73,122,89,126)(78,114,106,116)(80,94,81,92)(85,127,118,123)(87,120,97,125);(1,5,9,32)(2,12,15,56)(3,19)(4,24,28,82)(6,30,7,31)(8,33,36,91)(10,44)(11,48,50,108)(13,53,14,54)(16,55,59,113)(17,67)(18,69)(20,73)(21,29)(22,74)(23,78,49,106)(25,80,26,81)(27,83,42,95)(34,87)(35,90,39,93)(37,92,40,94)(38,97)(41,88)(43,98)(45,101)(46,52)(47,103)(51,107,65,117)(57,85)(58,112,62,115)(60,114,63,116)(61,118)(64,110)(66,119)(68,111)(70,120)(71,79)(72,121)(75,122)(76,89)(77,86)(84,126)(96,125)(99,127)(100,124)(102,105)(104,123)(109,128)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406859);