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#72340950

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#2840
commutator, Frattini, cosocleC25C25
prime radicalC22×#560C22
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

2910 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 2910 conjugacy classes of subgroups — too many to draw. Only its 491 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 — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gj,d^2=fgij,e^2=fgj,b^a=bj,c^a=ci,c^b=cgi,d^a=dij,d^b=dh,d^c=df,e^a=ej,e^b=efh,e^c=ehij,e^d=efg,f^b=fj,f^c=fi,f^d=fi,f^e=fij,g^d=gi,g^e=gij,h^b=hi,h^c=hij,h^d=hj,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72340952
maximal subgroup of#72340952, #72341033
maximal normal subgroup of#72340952, #72341033

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