Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #153078 |
| commutator, Frattini, cosocle | C22×C32 | C23 |
| LCS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 83 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 — 3 generators on 128 points:
(1,2)(3,16)(4,11)(5,28)(6,36)(7,41)(8,24)(9,15)(10,19)(12,32)(13,31)(14,38)(17,60)(18,57)(20,58)(21,69)(22,71)(23,64)(25,75)(26,80)(27,55)(29,82)(30,52)(33,99)(34,102)(35,85)(37,50)(39,106)(40,87)(42,59)(43,74)(44,65)(45,93)(46,92)(47,56)(48,78)(49,77)(51,113)(53,96)(54,66)(61,118)(62,114)(63,116)(67,121)(68,119)(70,122)(72,98)(73,95)(76,111)(79,125)(81,89)(83,94)(84,97)(86,105)(88,124)(90,103)(91,115)(100,109)(101,128)(104,112)(107,117)(108,126)(110,120)(123,127);(1,2)(3,17)(4,24)(5,29)(6,13)(7,14)(8,11)(9,15)(10,18)(12,35)(16,56)(19,64)(20,47)(21,61)(22,54)(23,44)(25,48)(26,73)(27,74)(28,82)(30,89)(31,36)(32,85)(33,88)(34,72)(37,103)(38,41)(39,49)(40,53)(42,107)(43,55)(45,62)(46,63)(50,76)(51,101)(52,104)(57,65)(58,60)(59,92)(66,119)(67,109)(68,110)(69,100)(70,108)(71,120)(75,78)(77,106)(79,83)(80,95)(81,115)(84,127)(86,90)(87,96)(91,112)(93,122)(94,125)(97,123)(98,102)(99,124)(105,111)(113,128)(114,126)(116,117)(118,121);(1,3)(2,10)(4,23)(5,30)(6,37)(7,42)(8,18)(9,20)(11,47)(12,50)(13,52)(14,54)(15,44)(16,28)(17,24)(19,32)(21,31)(22,38)(25,76)(26,81)(27,68)(29,86)(33,100)(34,90)(35,91)(36,45)(39,104)(40,105)(41,46)(43,63)(48,89)(49,111)(51,114)(53,115)(55,110)(56,78)(57,102)(58,85)(59,99)(60,98)(61,106)(62,95)(64,75)(65,82)(66,113)(67,96)(69,83)(70,77)(71,84)(72,112)(73,103)(74,107)(79,126)(80,109)(87,108)(88,122)(92,94)(93,97)(101,118)(116,128)(117,127)(119,125)(120,124)(121,123)Polycyclic presentation:
2^10|d^2=j,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=ce,c^b=cd,d^b=dj,d^c=dj,e^a=eg,e^b=ej,e^c=eg,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72322063, #72322943, #72322982 |
| maximal normal subgroup of | #72322063, #72322943, #72322982 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);