Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24354 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62637 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,15,20,9)(4,21)(5,27,30,11)(6,12)(7,13)(8,14)(10,24)(16,55,60,38)(17,39)(18,57,67,40)(19,53,36,41)(22,69,73,42)(23,70,74,43)(25,71)(26,54)(28,78,84,46)(29,79,82,47)(31,48)(32,49)(33,89,87,50)(34,90)(35,52)(37,45)(44,58)(51,88)(56,62)(59,107)(61,97,68,92)(63,93)(64,94)(65,95)(66,101,77,96)(72,116)(75,117,109,99)(76,106,86,100)(80,102)(81,121,124,103)(83,122,113,104)(85,105)(91,110)(98,118)(108,112)(111,127)(114,128)(115,125)(119,126)(120,123);(1,3,6,20)(2,9,12,15)(4,22,24,73)(5,17,30,62)(7,33,31,87)(8,36,32,19)(10,42,21,69)(11,39,27,56)(13,50,48,89)(14,53,49,41)(16,58,60,25)(18,65,67,35)(23,72,74,118)(26,77,37,66)(28,80,84,123)(29,85,82,64)(34,68,88,61)(38,71,55,44)(40,95,57,52)(43,98,70,116)(45,101,54,96)(46,102,78,120)(47,105,79,94)(51,97,90,92)(59,109,110,75)(63,113,112,83)(76,119,86,114)(81,115,124,111)(91,117,107,99)(93,122,108,104)(100,128,106,126)(103,127,121,125);(1,4)(2,10)(3,16)(5,23)(6,24)(7,34)(8,37)(9,38)(11,43)(12,21)(13,51)(14,54)(15,55)(17,59)(18,66)(19,68)(20,60)(22,33)(25,35)(26,32)(27,70)(28,81)(29,86)(30,74)(31,88)(36,61)(39,91)(40,96)(41,97)(42,50)(44,52)(45,49)(46,103)(47,106)(48,90)(53,92)(56,107)(57,101)(58,65)(62,110)(63,114)(64,115)(67,77)(69,89)(71,95)(72,80)(73,87)(75,83)(76,82)(78,121)(79,100)(84,124)(85,111)(93,126)(94,127)(98,102)(99,104)(105,125)(108,128)(109,113)(112,119)(116,120)(117,122)(118,123);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,39)(10,43)(12,27)(13,46)(14,47)(15,56)(16,59)(18,63)(19,64)(20,62)(21,70)(22,72)(24,74)(25,75)(26,76)(31,84)(32,82)(33,80)(34,81)(35,83)(36,85)(37,86)(38,91)(40,93)(41,94)(42,98)(44,99)(45,100)(48,78)(49,79)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(57,108)(58,109)(60,110)(61,111)(65,113)(66,114)(67,112)(68,115)(69,116)(71,117)(73,118)(77,119)(87,123)(88,124)(89,120)(90,121)(92,125)(95,122)(96,126)(97,127)(101,128)Polycyclic presentation:
2^9|a^2=fi,b^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408219);