Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24909 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,14)(3,15,20,64)(4,22,28,51)(5,29,34,97)(6,35,40,95)(7,41,38,13)(9,42,46,18)(10,27,52,26)(11,53,57,31)(12,58,61,39)(16,66,23,79)(17,44,71,107)(19,65,21,59)(24,82,86,127)(25,50,91,113)(30,98,75,54)(32,60,33,96)(36,47,73,45)(37,55,100,56)(43,102,48,87)(49,110,112,88)(62,74,109,69)(63,103,118,70)(67,111,80,116)(68,106,81,76)(72,93,104,117)(77,115,114,84)(78,89,101,90)(83,105,99,108)(85,94,92,123)(119,122,128,126)(120,125,124,121);(1,3,7,19,8,20,38,21)(2,9,13,45,14,46,41,47)(4,23,26,74,28,16,27,69)(5,30,32,71,34,75,33,17)(6,36,39,42,40,73,58,18)(10,48,22,76,52,43,51,106)(11,54,55,107,57,98,56,44)(12,59,35,64,61,65,95,15)(24,83,85,80,86,99,92,67)(25,87,89,81,91,102,90,68)(29,93,96,63,97,117,60,118)(31,70,100,72,53,103,37,104)(49,111,77,108,112,116,114,105)(50,66,78,109,113,79,101,62)(82,119,123,120,127,128,94,124)(84,125,88,122,115,121,110,126);(1,4)(2,10)(3,16)(5,24)(6,25)(7,27)(8,28)(9,43)(11,49)(12,50)(13,51)(14,52)(15,62)(17,67)(18,68)(19,74)(20,23)(21,69)(22,41)(26,38)(29,94)(30,99)(31,84)(32,92)(33,85)(34,86)(35,101)(36,102)(37,88)(39,90)(40,91)(42,81)(44,105)(45,76)(46,48)(47,106)(53,115)(54,116)(55,114)(56,77)(57,112)(58,89)(59,79)(60,82)(61,113)(63,119)(64,109)(65,66)(70,121)(71,80)(72,122)(73,87)(75,83)(78,95)(93,120)(96,127)(97,123)(98,111)(100,110)(103,125)(104,126)(107,108)(117,124)(118,128);(1,5,8,34)(2,11,14,57)(3,17,20,71)(4,24,28,86)(6,37,40,100)(7,32,38,33)(9,44,46,107)(10,49,52,112)(12,60,61,96)(13,55,41,56)(15,63,64,118)(16,67,23,80)(18,72,42,104)(19,30,21,75)(22,77,51,114)(25,88,91,110)(26,85,27,92)(29,95,97,35)(31,58,53,39)(36,103,73,70)(43,105,48,108)(45,54,47,98)(50,82,113,127)(59,117,65,93)(62,119,109,128)(66,120,79,124)(68,122,81,126)(69,83,74,99)(76,116,106,111)(78,123,101,94)(84,89,115,90)(87,121,102,125)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=hi,f^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,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=eh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409063);