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#72321950

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151115
commutatorC2×C32C22×C4
Frattini, cosocle#1397C23
prime radical#2262C2×C4
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1311 ◁ #7330 ◁ G
LECSG ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

153 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=eij,d^2=fhij,e^2=g,f^2=hij,h^2=ij,i^2=j,b^a=bgij,c^a=cd,c^b=ch,d^a=df,d^b=di,d^c=dfij,e^c=eij,e^d=ej,f^a=fh,f^b=fj,f^c=fhj,g^c=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72321955, #72321964, #72321969
maximal normal subgroup of#72321955, #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, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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);