Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25032 |
| 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 | |
465 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)(3,15,21,56)(4,22,28,45)(5,29,32,87)(6,33,35,91)(7,36,34,13)(9,37,41,19)(10,27,46,26)(11,47,49,30)(12,50,51,16)(17,60,23,74)(18,65,68,97)(20,70,69,55)(24,78,82,124)(25,44,83,105)(31,89,88,86)(38,94,42,99)(39,67,98,118)(40,57,54,93)(43,102,104,121)(48,59,85,107)(52,64,101,63)(53,92,110,66)(58,73,75,90)(61,100,76,95)(62,96,77,71)(72,106,103,79)(80,84,81,123)(108,127,125,128)(109,111,117,126)(112,122,120,113)(114,116,115,119);(1,3,16,57,7,20,35,37,8,21,50,93,34,69,6,19)(2,9,33,55,13,40,51,56,14,41,91,70,36,54,12,15)(4,23,75,94,27,63,25,77,28,17,58,99,26,64,83,62)(5,18,59,111,31,67,47,92,32,68,107,126,88,118,30,66)(10,42,73,74,45,96,44,101,46,38,90,60,22,71,105,52)(11,39,86,109,48,97,87,110,49,98,89,117,85,65,29,53)(24,76,121,127,81,114,79,122,82,61,102,128,80,115,106,113)(43,100,78,120,103,116,84,125,104,95,124,112,72,119,123,108);(1,4)(2,10)(3,17)(5,24)(6,25)(7,27)(8,28)(9,38)(11,43)(12,44)(13,45)(14,46)(15,52)(16,58)(18,61)(19,62)(20,64)(21,23)(22,36)(26,34)(29,84)(30,79)(31,81)(32,82)(33,90)(35,83)(37,77)(39,95)(40,71)(41,42)(47,106)(48,103)(49,104)(50,75)(51,105)(53,108)(54,96)(55,74)(56,101)(57,94)(59,102)(60,70)(63,69)(65,116)(66,113)(67,115)(68,76)(72,85)(73,91)(78,89)(80,88)(86,124)(87,123)(92,122)(93,99)(97,119)(98,100)(107,121)(109,120)(110,125)(111,127)(112,117)(114,118)(126,128);(1,5,16,59,7,31,35,47,8,32,50,107,34,88,6,30)(2,11,33,86,13,48,51,87,14,49,91,89,36,85,12,29)(3,18,57,111,20,67,37,92,21,68,93,126,69,118,19,66)(4,24,58,102,27,81,83,106,28,82,75,121,26,80,25,79)(9,39,55,109,40,97,56,110,41,98,70,117,54,65,15,53)(10,43,90,124,45,103,105,123,46,104,73,78,22,72,44,84)(17,61,94,127,64,115,77,122,23,76,99,128,63,114,62,113)(38,95,74,120,71,119,101,125,42,100,60,112,96,116,52,108)Polycyclic presentation:
2^9|a^2=i,b^2=ef,d^2=eh,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,g^b=gi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409193);