Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24267 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,16)(3,17,24,61)(4,25,30,49)(5,31,35,91)(6,36,38,55)(7,15,8,14)(10,40,45,21)(11,39,50,29)(12,51,54,32)(13,37,56,18)(19,65,70,111)(20,71,75,98)(22,60,23,59)(26,80,69,57)(27,83,86,125)(28,87,66,48)(33,90,34,89)(41,82,77,112)(42,74,99,73)(43,76,44,62)(46,95,100,68)(47,102,104,114)(52,92,53,64)(58,96,110,72)(63,79,94,105)(67,97,118,121)(78,106,103,84)(81,101,117,119)(85,88,93,124)(107,123,122,127)(108,120,109,113)(115,126,128,116);(1,3,18,62,8,23,38,40,9,24,37,76,7,22,6,21)(2,10,36,60,15,44,56,61,16,45,55,59,14,43,13,17)(4,26,63,68,29,19,66,112,30,69,94,95,39,70,28,82)(5,20,64,113,34,74,51,96,35,75,92,120,33,73,32,72)(11,46,79,80,25,77,87,111,50,100,105,57,49,41,48,65)(12,42,90,109,53,98,91,110,54,99,89,108,52,71,31,58)(27,81,114,116,85,67,106,127,86,117,102,126,93,118,84,123)(47,101,83,122,78,121,124,128,104,119,125,107,103,97,88,115);(1,4)(2,11)(3,19)(5,27)(6,28)(7,39)(8,29)(9,30)(10,41)(12,47)(13,48)(14,49)(15,25)(16,50)(17,57)(18,63)(20,67)(21,68)(22,26)(23,69)(24,70)(31,88)(32,84)(33,93)(34,85)(35,86)(36,79)(37,94)(38,66)(40,95)(42,97)(43,100)(44,46)(45,77)(51,106)(52,103)(53,78)(54,104)(55,105)(56,87)(58,107)(59,111)(60,65)(61,80)(62,112)(64,114)(71,119)(72,116)(73,81)(74,117)(75,118)(76,82)(83,90)(89,125)(91,124)(92,102)(96,126)(98,101)(99,121)(108,128)(109,115)(110,122)(113,127)(120,123);(1,5,18,64,8,34,38,51,9,35,37,92,7,33,6,32)(2,12,36,90,15,53,56,91,16,54,55,89,14,52,13,31)(3,20,62,113,23,74,40,96,24,75,76,120,22,73,21,72)(4,27,63,114,29,85,66,106,30,86,94,102,39,93,28,84)(10,42,60,109,44,98,61,110,45,99,59,108,43,71,17,58)(11,47,79,83,25,78,87,124,50,104,105,125,49,103,48,88)(19,67,112,127,69,117,95,126,70,118,82,123,26,81,68,116)(41,97,65,115,46,101,80,122,77,121,111,128,100,119,57,107)Polycyclic presentation:
2^9|a^2=i,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,f^a=fi,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, 408057);