Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509179 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30444 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33538 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,12,38,13,61,58,8,15,40,60,7,14,39,59)(3,16,19,62,20,79,77,44,22,65,78,43,21,56,36,9)(4,23,26,82,27,10,45,48,29,49,94,112,28,50,93,113)(5,30,32,55,34,96,100,54,35,57,72,11,33,52,37,53)(17,66,68,80,24,83,85,103,70,117,121,41,69,63,101,46)(18,71,31,64,73,42,98,51,75,104,99,115,74,105,76,116)(25,86,88,81,89,95,119,110,91,126,106,109,90,111,92,47)(67,102,97,114,122,128,127,124,120,123,125,118,84,107,87,108);(1,3,8,22)(2,9,15,44)(4,24,29,69)(5,31,35,99)(6,36,40,77)(7,20,38,21)(10,46,50,103)(11,51,55,116)(12,56,60,79)(13,43,14,62)(16,58,65,59)(17,28,70,27)(18,72,75,32)(19,61,78,39)(23,80,49,41)(25,87,91,127)(26,68,94,121)(30,71,57,104)(33,98,34,76)(37,73,100,74)(42,96,105,52)(45,101,93,85)(47,108,110,124)(48,63,113,83)(53,64,54,115)(66,112,117,82)(67,119,120,92)(81,118,109,114)(84,106,122,88)(86,107,126,128)(89,125,90,97)(95,123,111,102);(1,4)(2,10)(3,17)(5,25)(6,26)(7,28)(8,29)(9,41)(11,47)(12,48)(13,49)(14,23)(15,50)(16,63)(18,67)(19,68)(20,24)(21,69)(22,70)(27,38)(30,95)(31,97)(32,88)(33,90)(34,89)(35,91)(36,101)(37,92)(39,93)(40,94)(42,102)(43,103)(44,80)(45,61)(46,62)(51,114)(52,86)(53,81)(54,109)(55,110)(56,117)(57,111)(58,112)(59,82)(60,113)(64,108)(65,83)(66,79)(71,107)(72,106)(73,122)(74,84)(75,120)(76,87)(77,85)(78,121)(96,126)(98,127)(99,125)(100,119)(104,128)(105,123)(115,124)(116,118);(1,5,8,35)(2,11,15,55)(3,18,22,75)(4,25,29,91)(6,37,40,100)(7,34,38,33)(9,42,44,105)(10,47,50,110)(12,57,60,30)(13,54,14,53)(16,64,65,115)(17,67,70,120)(19,76,78,98)(20,74,21,73)(23,81,49,109)(24,84,69,122)(26,92,94,119)(27,90,28,89)(31,77,99,36)(32,61,72,39)(41,102,80,123)(43,104,62,71)(45,106,93,88)(46,107,103,128)(48,111,113,95)(51,79,116,56)(52,58,96,59)(63,108,83,124)(66,118,117,114)(68,87,121,127)(82,86,112,126)(85,125,101,97)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416378);