Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31851 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,14,9,13,8,16)(3,10,20,43,19,46,22,45)(4,23,27,51,26,50,29,11)(5,12,31,53,33,52,34,56)(7,15,35,57,40,55,39,61)(17,62,65,98,67,99,66,41)(18,68,69,102,30,42,72,100)(21,44,73,103,71,105,36,58)(24,79,81,107,82,76,74,47)(25,83,85,110,84,77,88,48)(28,78,63,112,87,59,37,49)(32,54,91,113,38,60,94,116)(64,119,120,97,90,128,122,123)(70,104,96,117,75,101,92,114)(80,127,125,106,121,126,124,108)(86,115,93,111,89,118,95,109);(1,3,8,20,9,19,6,22)(2,10,14,45,13,46,16,43)(4,24,27,74,26,82,29,81)(5,30,34,72,33,18,31,69)(7,36,39,21,40,73,35,71)(11,47,50,79,51,107,23,76)(12,42,53,102,52,68,56,100)(15,58,57,105,55,103,61,44)(17,63,66,87,67,37,65,28)(25,80,85,124,84,121,88,125)(32,92,94,70,38,96,91,75)(41,59,62,112,98,78,99,49)(48,108,77,127,110,106,83,126)(54,114,113,101,60,117,116,104)(64,95,122,86,90,93,120,89)(97,118,128,111,123,115,119,109);(1,4,9,26)(2,11,13,51)(3,17,19,67)(5,25,33,84)(6,29,8,27)(7,37,40,63)(10,41,46,98)(12,48,52,110)(14,50,16,23)(15,59,55,78)(18,64,30,90)(20,66,22,65)(21,74,71,81)(24,73,82,36)(28,35,87,39)(31,88,34,85)(32,93,38,95)(42,97,68,123)(43,99,45,62)(44,76,105,79)(47,103,107,58)(49,57,112,61)(53,77,56,83)(54,115,60,118)(69,122,72,120)(70,124,75,125)(80,96,121,92)(86,94,89,91)(100,119,102,128)(101,127,104,126)(106,114,108,117)(109,116,111,113);(1,5,8,31,9,33,6,34)(2,12,14,56,13,52,16,53)(3,18,20,72,19,30,22,69)(4,25,27,88,26,84,29,85)(7,38,39,94,40,32,35,91)(10,42,45,100,46,68,43,102)(11,48,50,83,51,110,23,77)(15,60,57,113,55,54,61,116)(17,64,66,120,67,90,65,122)(21,75,73,96,71,70,36,92)(24,80,74,125,82,121,81,124)(28,89,63,93,87,86,37,95)(41,97,62,119,98,123,99,128)(44,104,58,117,105,101,103,114)(47,106,79,127,107,108,76,126)(49,111,59,118,112,109,78,115)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,d^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417903);