Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511767 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32878 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #63093 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3051 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)(3,17,22,58)(4,23,27,53)(5,12,30,49)(6,31,35,51)(7,15,8,14)(10,39,43,20)(11,36,46,26)(13,33,52,34)(18,59,63,99)(19,57,68,107)(21,69,38,50)(24,73,76,71)(25,78,81,45)(28,82,85,103)(29,54,37,48)(32,55,86,42)(40,79,92,113)(41,90,96,65)(44,77,98,75)(47,88,101,84)(56,64,105,62)(60,109,112,108)(61,70,89,93)(66,100,67,106)(72,87,102,80)(74,97,122,124)(83,94,115,95)(91,126,127,120)(104,114,128,111)(110,118,125,116)(117,123,119,121);(1,3)(2,10)(4,18)(5,19)(6,32)(7,21)(8,38)(9,22)(11,40)(12,41)(13,50)(14,42)(15,55)(16,43)(17,31)(20,34)(23,70)(24,60)(25,79)(26,62)(27,63)(28,83)(29,66)(30,68)(33,39)(35,86)(36,64)(37,67)(44,91)(45,99)(46,92)(47,100)(48,94)(49,96)(51,58)(52,69)(53,93)(54,95)(56,72)(57,82)(59,78)(61,80)(65,84)(71,116)(73,118)(74,120)(75,111)(76,112)(77,114)(81,113)(85,115)(87,89)(88,90)(97,109)(98,127)(101,106)(102,105)(103,107)(104,117)(108,124)(110,123)(119,128)(121,125)(122,126);(1,4)(2,11)(3,18)(5,24)(6,25)(7,36)(8,26)(9,27)(10,40)(12,44)(13,45)(14,53)(15,23)(16,46)(17,56)(19,60)(20,61)(21,64)(22,63)(28,74)(29,77)(30,76)(31,72)(32,79)(33,87)(34,80)(35,81)(37,75)(38,62)(39,89)(41,91)(42,93)(43,92)(47,97)(48,71)(49,98)(50,99)(51,102)(52,78)(54,73)(55,70)(57,104)(58,105)(59,69)(65,110)(66,114)(67,111)(68,112)(82,117)(83,120)(84,123)(85,122)(86,113)(88,121)(90,125)(94,116)(95,118)(96,127)(100,109)(101,124)(103,119)(106,108)(107,128)(115,126);(1,5,9,30)(2,12,16,49)(3,19,22,68)(4,24,27,76)(6,28,35,85)(7,37,8,29)(10,41,43,96)(11,44,46,98)(13,47,52,101)(14,54,15,48)(17,57,58,107)(18,60,63,112)(20,65,39,90)(21,67,38,66)(23,71,53,73)(25,74,81,122)(26,77,36,75)(31,82,51,103)(32,83,86,115)(33,88,34,84)(40,91,92,127)(42,95,55,94)(45,97,78,124)(50,100,69,106)(56,104,105,128)(59,108,99,109)(61,110,89,125)(62,114,64,111)(70,116,93,118)(72,117,102,119)(79,120,113,126)(80,123,87,121)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,f^c=fi,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418966);