Finite Group Databasebeta

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#72321773

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#513003
commutator, Frattini, cosocleC2×C32C24
prime radical#153607C2
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ C4×M64 ◁ G
LECSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

426 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(1,2)(3,15)(4,22)(5,29)(6,34)(7,38)(8,10)(9,18)(11,26)(12,25)(13,31)(14,36)(16,54)(17,56)(19,65)(20,67)(21,53)(23,68)(24,49)(27,85)(28,87)(30,46)(32,96)(33,89)(35,55)(37,97)(39,57)(40,60)(41,79)(42,78)(43,74)(44,73)(45,95)(47,108)(48,110)(50,113)(51,64)(52,63)(58,118)(59,115)(61,109)(62,116)(66,120)(69,80)(70,83)(71,76)(72,104)(75,126)(77,114)(81,91)(82,90)(84,98)(86,112)(88,111)(92,102)(93,107)(94,106)(99,123)(100,119)(101,127)(103,128)(105,117)(121,124)(122,125);(3,16)(4,23)(9,39)(11,43)(17,21)(19,58)(20,59)(22,68)(26,74)(27,75)(28,76)(30,92)(35,62)(40,52)(41,99)(42,100)(47,103)(48,98)(49,104)(51,105)(61,66)(69,121)(70,122)(71,87)(80,124)(83,125)(84,110)(85,126)(93,94)(101,112)(108,128)(111,114);(1,3)(2,9)(4,24)(5,30)(6,35)(7,17)(8,16)(10,39)(11,46)(12,49)(13,51)(14,52)(15,22)(18,26)(19,25)(20,31)(21,36)(23,72)(27,86)(28,88)(29,41)(32,93)(33,92)(34,42)(37,62)(38,40)(43,102)(44,104)(45,105)(47,109)(48,106)(50,114)(53,71)(54,68)(55,85)(56,87)(57,74)(58,73)(59,95)(60,84)(61,82)(63,110)(64,108)(65,69)(66,113)(67,70)(75,127)(76,77)(78,80)(79,83)(81,94)(89,99)(90,111)(91,112)(96,101)(97,100)(98,107)(103,120)(115,122)(116,126)(117,128)(118,121)(119,124)(123,125);(2,10)(3,17,16,21)(4,23)(9,40,39,52)(12,44)(13,45)(14,38)(15,53,54,56)(18,63,57,60)(19,61,58,66)(20,62,59,35)(24,77,72,88)(26,74)(27,75)(28,76)(29,89)(30,93,92,94)(34,97)(41,101,99,112)(42,51,100,105)(46,106,102,107)(49,111,104,114)(50,90)(55,115,116,67)(64,78,117,119)(65,120,118,109)(79,86,123,127)(80,124)(83,125)(84,110)(91,96)(108,128)

Polycyclic presentation:

2^10|d^2=f,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=ce,c^b=cj,d^a=dj,d^c=di,e^a=eg,e^b=ej,e^c=eg,e^d=ej,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321842, #72321964, #72321969
maximal normal subgroup of#72321842, #72321964, #72321969

GAP

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.6);;
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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
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, 5, 4, F.5*F.10);;
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);