Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31000 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
645 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)(3,16)(4,11)(5,28,31,12)(6,33)(7,14)(8,13)(9,15)(10,19)(17,60)(18,51,66,61)(20,57)(21,44)(22,55)(23,73)(24,46,78,82)(25,49)(26,48)(27,50)(29,42,68,98)(30,52,34,54)(32,83,87,53)(35,90)(36,92)(37,58)(38,56)(39,59)(40,47)(41,64)(43,69)(45,70)(62,94)(63,115,117,109)(65,96)(67,110,88,114)(71,101)(72,97)(74,121,122,104)(75,100)(76,123,119,102)(77,107,80,105)(79,106,116,125)(81,103)(84,128,118,95)(85,99,120,126)(86,111,91,113)(89,112,124,127)(93,108);(1,3,9,22)(2,10,15,44)(4,23,27,75)(5,29,31,68)(6,21,8,19)(7,36,39,69)(11,45,50,103)(12,51,28,61)(13,55,33,16)(14,57,59,101)(17,62,65,26)(18,34,66,30)(20,35,71,38)(24,76,78,119)(25,81,40,70)(32,88,87,67)(37,72,93,64)(41,94,97,48)(42,54,98,52)(43,90,92,56)(46,104,82,121)(47,100,49,73)(53,99,83,126)(58,60,108,96)(63,89,117,124)(74,80,122,77)(79,84,116,118)(85,91,120,86)(95,127,128,112)(102,107,123,105)(106,109,125,115)(110,111,114,113);(1,4,9,27)(2,11,15,50)(3,17,22,65)(5,24,31,78)(6,25,8,40)(7,37,39,93)(10,41,44,97)(12,46,28,82)(13,47,33,49)(14,58,59,108)(16,60,55,96)(18,63,66,117)(19,64,21,72)(20,70,71,81)(23,69,75,36)(26,35,62,38)(29,84,68,118)(30,77,34,80)(32,89,87,124)(42,95,98,128)(43,100,92,73)(45,101,103,57)(48,90,94,56)(51,109,61,115)(52,105,54,107)(53,112,83,127)(67,119,88,76)(74,120,122,85)(79,91,116,86)(99,104,126,121)(102,110,123,114)(106,111,125,113);(1,5)(2,12)(3,18)(4,24)(6,34)(7,32)(8,30)(9,31)(10,42)(11,46)(13,54)(14,53)(15,28)(16,61)(17,63)(19,68)(20,67)(21,29)(22,66)(23,74)(25,80)(26,79)(27,78)(33,52)(35,91)(36,85)(37,89)(38,86)(39,87)(40,77)(41,95)(43,99)(44,98)(45,102)(47,107)(48,106)(49,105)(50,82)(51,55)(56,113)(57,114)(58,112)(59,83)(60,115)(62,116)(64,118)(65,117)(69,120)(70,119)(71,88)(72,84)(73,121)(75,122)(76,81)(90,111)(92,126)(93,124)(94,125)(96,109)(97,128)(100,104)(101,110)(103,123)(108,127)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416978);