Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499781 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1229 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)(3,17,23,66)(4,11,28,52)(5,29,31,90)(6,33,39,59)(7,15,8,14)(10,42,47,20)(12,53,54,30)(13,37,60,38)(18,63,69,100)(19,44,71,103)(21,56,22,65)(24,78,76,112)(25,81,83,89)(26,86,35,93)(27,61,40,51)(32,92,41,58)(34,45,75,46)(36,62,97,55)(43,68,101,73)(48,94,105,80)(49,91,108,82)(50,96,57,88)(64,115,116,70)(67,118,120,99)(72,106,77,104)(74,102,98,117)(79,107,121,127)(84,111,85,113)(87,109,95,110)(114,123,128,119)(122,126,124,125);(1,3,9,23)(2,10,16,47)(4,24,28,76)(5,19,31,71)(6,34,39,75)(7,22,8,21)(11,48,52,105)(12,44,54,103)(13,56,60,65)(14,46,15,45)(17,59,66,33)(18,40,69,27)(20,37,42,38)(25,79,83,121)(26,73,35,68)(29,64,90,116)(30,74,53,98)(32,77,41,72)(36,70,97,115)(43,61,101,51)(49,107,108,127)(50,63,57,100)(55,106,62,104)(58,102,92,117)(67,85,120,84)(78,93,112,86)(80,96,94,88)(81,125,89,126)(82,119,91,123)(87,124,95,122)(99,110,118,109)(111,114,113,128);(1,4)(2,11)(3,18)(5,25)(6,35)(7,40)(8,27)(9,28)(10,43)(12,49)(13,57)(14,61)(15,51)(16,52)(17,63)(19,67)(20,73)(21,76)(22,24)(23,69)(26,39)(29,89)(30,91)(31,83)(32,85)(33,93)(34,94)(36,95)(37,88)(38,96)(41,84)(42,68)(44,99)(45,105)(46,48)(47,101)(50,60)(53,82)(54,108)(55,110)(56,112)(58,113)(59,86)(62,109)(64,114)(65,78)(66,100)(70,123)(71,120)(72,121)(74,124)(75,80)(77,79)(81,90)(87,97)(92,111)(98,122)(102,126)(103,118)(104,127)(106,107)(115,119)(116,128)(117,125);(1,5)(2,12)(3,19)(4,25)(6,36)(7,32)(8,41)(9,31)(10,44)(11,49)(13,58)(14,55)(15,62)(16,54)(17,64)(18,67)(20,74)(21,72)(22,77)(23,71)(24,79)(26,87)(27,84)(28,83)(29,33)(30,38)(34,70)(35,95)(37,53)(39,97)(40,85)(42,98)(43,99)(45,104)(46,106)(47,103)(48,107)(50,111)(51,109)(52,108)(56,102)(57,113)(59,90)(60,92)(61,110)(63,114)(65,117)(66,116)(68,122)(69,120)(73,124)(75,115)(76,121)(78,125)(80,119)(81,86)(82,88)(89,93)(91,96)(94,123)(100,128)(101,118)(105,127)(112,126)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=eh,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, 406980);