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#72341227

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#238077
commutator, Frattini, cosocleC23×D4C24
prime radicalC22×H(5,𝔽2)C23
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

1777 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1777 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,17,40)(4,9,21,36)(5,10,24,39)(6,25,29,11)(8,26,32,16)(14,43,18,48)(15,44,22,54)(19,51,53,34)(20,41,27,35)(23,42,28,38)(30,50,33,46)(31,55,37,47)(45,61,49,64)(52,59,56,60)(57,62,58,63);(1,3,6,26)(2,8,11,40)(4,18,20,46)(5,22,28,55)(7,17,29,16)(9,33,35,43)(10,37,42,54)(12,32,25,13)(14,27,50,21)(15,23,47,24)(19,45,56,62)(30,41,48,36)(31,38,44,39)(34,57,60,64)(49,52,63,53)(51,58,59,61);(1,4)(2,9)(3,14)(5,19,24,53)(6,27,29,20)(7,21)(8,30)(10,34,39,51)(11,41,25,35)(12,36)(13,43,40,48)(15,45,22,49)(16,46,26,50)(17,18)(23,52)(28,56)(31,57,37,58)(32,33)(38,59)(42,60)(44,61)(47,63)(54,64)(55,62);(1,5)(2,10)(3,15)(4,19,21,53)(6,28,29,23)(7,24)(8,31)(9,34,36,51)(11,42,25,38)(12,39)(13,44,40,54)(14,45,18,49)(16,47,26,55)(17,22)(20,52)(27,56)(30,57,33,58)(32,37)(35,59)(41,60)(43,61)(46,63)(48,64)(50,62)

Polycyclic presentation:

2^10|a^2=j,b^2=eg,c^2=gij,d^2=ghj,e^2=j,b^a=be,c^a=cf,c^b=cfghj,d^a=dfhi,d^b=dfghi,d^c=dhi,e^a=ej,e^b=ei,e^c=eh,e^d=eh,f^b=fhij,f^c=fi,f^d=fij,g^a=gi,g^b=gi,g^d=gj,g^e=gj,h^b=hj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341228, #72341239
maximal normal subgroup of#72341228, #72341239

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.10);;
gap> SetPower(coll, 2, F.5*F.7);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);