Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510804 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31945 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
409 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)(3,17)(4,24,29,49)(5,12,33,56)(6,15)(7,13)(8,14)(9,16)(10,20)(11,27,51,34)(18,42,67,101)(19,68,71,52)(21,58)(22,46)(23,44)(25,47,81,107)(26,82)(28,78,90,59)(30,91,69,43)(31,54,36,53)(32,55,40,60)(35,61)(37,57)(38,97)(39,50,88,112)(41,62)(45,72)(48,84)(63,66,100,65)(64,99)(70,118,74,114)(73,76,80,106)(75,103)(77,87)(79,105)(83,126)(85,110)(86,117)(89,119)(92,123)(93,104,124,102)(94,116,96,115)(95,111)(98,109)(108,125)(113,120)(121,127)(122,128);(1,3,6,20,9,23,7,22)(2,10,13,17,16,46,15,44)(4,25,34,80,29,81,27,73)(5,30,36,19,33,69,31,71)(8,38,35,21,41,72,37,75)(11,47,24,76,51,107,49,106)(12,52,54,43,56,68,53,91)(14,58,57,97,62,103,61,45)(18,39,66,90,67,88,65,28)(26,83,84,121,87,125,85,79)(32,70,96,124,40,74,94,93)(42,78,63,112,101,59,100,50)(48,108,82,127,110,126,77,105)(55,104,116,114,60,102,115,118)(64,89,120,95,122,86,92,98)(99,111,123,119,128,109,113,117);(1,4,9,29)(2,11)(3,18)(5,26,33,87)(6,34,7,27)(8,39,41,88)(10,42,46,101)(12,48)(13,24)(14,59)(15,49)(16,51)(17,63,44,100)(19,64)(20,66)(21,73)(22,65)(23,67)(25,72)(28,35,90,37)(30,92)(31,85,36,84)(32,95,40,98)(38,81)(43,99,91,128)(45,76,97,106)(47,58,107,103)(50,57)(52,113,68,123)(53,77)(54,82)(55,117)(56,110)(60,119)(61,112)(62,78)(69,120)(70,125)(71,122)(74,83)(75,80)(79,124)(86,94,89,96)(93,121)(102,108,104,126)(105,118,127,114)(109,115)(111,116);(1,5)(2,12,16,56)(3,19,23,71)(4,26)(6,31)(7,36)(8,40)(9,33)(10,43)(11,48,51,110)(13,53,15,54)(14,60,62,55)(17,52)(18,64,67,122)(20,30,22,69)(21,74,75,70)(24,77,49,82)(25,79,81,121)(27,84)(28,89)(29,87)(32,41)(34,85)(35,96)(37,94)(38,93,72,124)(39,98)(42,99)(44,68)(45,104)(46,91)(47,105)(50,111,112,109)(57,116,61,115)(58,118)(59,119,78,117)(63,113)(65,120,66,92)(73,83,80,125)(76,126)(86,90)(88,95)(97,102)(100,123)(101,128)(103,114)(106,108)(107,127)Polycyclic presentation:
2^9|a^2=fh,b^2=efh,c^2=eh,d^2=ehi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=ghi,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418001);