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#72322771

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152629
commutatorC2×C32C22×C4
Frattini, cosocleC22×C32C23
prime radical#512941C2
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G
LECSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

389 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 128 points:

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

Polycyclic presentation:

2^10|c^2=degj,d^2=gh,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cd,c^b=cfj,d^a=dg,d^b=dj,d^c=dgj,e^a=egj,e^c=egj,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

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72322773, #72322916, #72322960
maximal normal subgroup of#72322773, #72322916, #72322960

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, 3, F.4*F.5*F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
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.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*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);