Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24424 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61108 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,7,14,9,15,8,40)(3,16,20,60,22,61,21,55)(4,23,27,51,28,50,39,11)(5,29,31,88,32,89,41,56)(6,33,36,57,38,92,37,13)(10,34,44,73,46,72,45,19)(12,35,52,90,54,93,53,30)(17,62,65,108,66,103,24,58)(18,43,68,99,70,101,69,100)(25,78,80,121,82,123,81,87)(26,49,85,75,86,105,94,91)(42,64,96,118,97,83,47,77)(48,79,74,120,104,124,106,84)(59,109,110,119,111,71,98,67)(63,113,115,126,116,102,76,95)(107,114,112,117,127,128,125,122);(1,3)(2,10)(4,24)(5,18)(6,34)(7,20)(8,21)(9,22)(11,47)(12,43)(13,55)(14,44)(15,46)(16,33)(17,27)(19,37)(23,42)(25,76)(26,83)(28,65)(29,59)(30,71)(31,68)(32,70)(35,67)(36,73)(38,72)(39,66)(40,45)(41,69)(48,102)(49,58)(50,97)(51,96)(52,99)(53,100)(54,101)(56,98)(57,60)(61,92)(62,75)(63,80)(64,86)(74,95)(77,85)(78,107)(79,114)(81,116)(82,115)(84,122)(87,125)(88,110)(89,111)(90,109)(91,103)(93,119)(94,118)(104,113)(105,108)(106,126)(112,121)(117,120)(123,127)(124,128);(1,4,9,28)(2,11,15,51)(3,17,22,66)(5,25,32,82)(6,26,38,86)(7,39,8,27)(10,42,46,97)(12,48,54,104)(13,49,57,105)(14,50,40,23)(16,58,61,108)(18,63,70,116)(19,64,73,83)(20,24,21,65)(29,87,89,121)(30,79,90,124)(31,81,41,80)(33,91,92,75)(34,77,72,118)(35,84,93,120)(36,94,37,85)(43,95,101,126)(44,47,45,96)(52,106,53,74)(55,62,60,103)(56,78,88,123)(59,107,111,127)(67,114,119,128)(68,76,69,115)(71,117,109,122)(98,112,110,125)(99,102,100,113);(1,5)(2,12)(3,18)(4,25)(6,35)(7,31)(8,41)(9,32)(10,43)(11,48)(13,56)(14,52)(15,54)(16,59)(17,63)(19,71)(20,68)(21,69)(22,70)(23,74)(24,76)(26,84)(27,80)(28,82)(29,33)(30,37)(34,67)(36,90)(38,93)(39,81)(40,53)(42,95)(44,99)(45,100)(46,101)(47,102)(49,78)(50,106)(51,104)(55,98)(57,88)(58,107)(60,110)(61,111)(62,112)(64,117)(65,115)(66,116)(72,119)(73,109)(75,121)(77,114)(79,85)(83,122)(86,120)(87,91)(89,92)(94,124)(96,113)(97,126)(103,125)(105,123)(108,127)(118,128)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408450);