Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501270 |
| 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:
(1,2,7,13,8,15,41,14)(3,16,20,54,21,63,40,62)(4,22,26,50,28,59,27,10)(5,29,31,56,33,92,32,91)(6,34,38,12,39,57,94,58)(9,35,44,19,46,75,45,76)(11,37,51,30,53,95,52,96)(17,64,23,79,68,108,67,60)(18,43,70,103,72,105,71,104)(24,82,84,109,86,112,85,90)(25,55,88,97,36,49,89,78)(42,66,47,81,101,73,106,98)(48,83,110,87,111,93,77,99)(61,74,114,69,115,123,102,124)(65,117,119,127,120,107,80,100)(113,118,128,121,126,122,116,125);(1,3,8,21)(2,9,15,46)(4,23,28,67)(5,18,33,72)(6,35,39,75)(7,40,41,20)(10,47,50,106)(11,43,53,105)(12,54,58,62)(13,45,14,44)(16,34,63,57)(17,27,68,26)(19,38,76,94)(22,42,59,101)(24,80,86,119)(25,73,36,66)(29,61,92,115)(30,74,96,123)(31,71,32,70)(37,69,95,124)(48,107,111,117)(49,79,55,60)(51,104,52,103)(56,102,91,114)(64,97,108,78)(65,84,120,85)(77,100,110,127)(81,89,98,88)(82,126,112,113)(83,118,93,122)(87,125,99,121)(90,116,109,128);(1,4,8,28)(2,10,15,50)(3,17,21,68)(5,24,33,86)(6,36,39,25)(7,27,41,26)(9,42,46,101)(11,48,53,111)(12,55,58,49)(13,59,14,22)(16,60,63,79)(18,65,72,120)(19,73,76,66)(20,67,40,23)(29,90,92,109)(30,93,96,83)(31,85,32,84)(34,97,57,78)(35,98,75,81)(37,99,95,87)(38,88,94,89)(43,100,105,127)(44,106,45,47)(51,77,52,110)(54,108,62,64)(56,112,91,82)(61,113,115,126)(69,122,124,118)(70,80,71,119)(74,125,123,121)(102,128,114,116)(103,107,104,117);(1,5)(2,11)(3,18)(4,24)(6,37)(7,32)(8,33)(9,43)(10,48)(12,56)(13,52)(14,51)(15,53)(16,61)(17,65)(19,74)(20,71)(21,72)(22,77)(23,80)(25,87)(26,85)(27,84)(28,86)(29,57)(30,94)(31,41)(34,92)(35,69)(36,99)(38,96)(39,95)(40,70)(42,100)(44,104)(45,103)(46,105)(47,107)(49,82)(50,111)(54,102)(55,112)(58,91)(59,110)(60,113)(62,114)(63,115)(64,116)(66,121)(67,119)(68,120)(73,125)(75,124)(76,123)(78,90)(79,126)(81,118)(83,88)(89,93)(97,109)(98,122)(101,127)(106,117)(108,128)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=h,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=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ehi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408469);