Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501701 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24780 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,15,9,16,7,14)(3,17,39,60,23,61,22,59)(4,24,29,11,30,49,28,50)(5,31,34,91,35,92,33,90)(6,36,37,56,38,55,18,13)(10,40,44,75,45,62,43,21)(12,51,53,93,54,64,52,32)(19,65,69,57,70,112,25,79)(20,71,94,42,74,98,73,99)(26,82,85,89,86,125,84,124)(27,87,63,78,66,48,88,106)(41,96,100,81,76,68,46,101)(47,103,104,107,105,83,77,114)(58,95,110,120,111,113,109,72)(67,97,117,119,118,121,80,102)(108,126,128,123,122,116,115,127);(1,3,18,62,7,22,38,75,9,23,37,40,8,39,6,21)(2,10,36,60,15,44,56,61,16,45,55,59,14,43,13,17)(4,25,63,68,29,19,66,101,30,69,88,96,28,70,27,81)(5,20,64,113,33,73,93,120,35,74,51,95,34,94,32,72)(11,46,78,79,24,76,87,112,50,100,106,57,49,41,48,65)(12,42,91,110,53,98,92,111,54,99,90,109,52,71,31,58)(26,80,114,116,85,67,103,127,86,117,107,126,84,118,83,123)(47,102,82,122,77,121,124,128,105,119,125,108,104,97,89,115);(1,4)(2,11)(3,19)(5,26)(6,27)(7,29)(8,28)(9,30)(10,41)(12,47)(13,48)(14,49)(15,24)(16,50)(17,57)(18,63)(20,67)(21,68)(22,69)(23,70)(25,39)(31,89)(32,83)(33,85)(34,84)(35,86)(36,78)(37,88)(38,66)(40,81)(42,97)(43,100)(44,46)(45,76)(51,107)(52,104)(53,77)(54,105)(55,106)(56,87)(58,108)(59,112)(60,65)(61,79)(62,101)(64,114)(71,119)(72,116)(73,117)(74,118)(75,96)(80,94)(82,91)(90,125)(92,124)(93,103)(95,123)(98,102)(99,121)(109,128)(110,115)(111,122)(113,127)(120,126);(1,5,18,64,7,33,38,93,9,35,37,51,8,34,6,32)(2,12,36,91,15,53,56,92,16,54,55,90,14,52,13,31)(3,20,62,113,22,73,75,120,23,74,40,95,39,94,21,72)(4,26,63,114,29,85,66,103,30,86,88,107,28,84,27,83)(10,42,60,110,44,98,61,111,45,99,59,109,43,71,17,58)(11,47,78,82,24,77,87,124,50,105,106,125,49,104,48,89)(19,67,101,127,69,117,96,126,70,118,81,123,25,80,68,116)(41,97,65,115,46,102,79,122,76,121,112,128,100,119,57,108)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408898);