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#72285872

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72235677
commutator, Frattini, cosocle#2309C24
prime radical#72283758C2
LCSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35616 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

4481 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 4481 conjugacy classes of subgroups — too many to draw. Only its 164 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)(3,9)(4,18)(5,24)(6,12)(7,13)(8,14)(10,21)(11,28)(15,35)(16,32)(17,33)(19,52)(20,31)(22,50)(23,51)(25,41)(26,49)(27,44)(29,40)(30,45)(34,57)(36,53)(37,58)(38,46)(39,56)(42,47)(43,59)(48,55)(54,62)(60,64)(61,63);(1,3)(2,9)(4,19)(5,25)(6,28)(7,16,30,47)(8,17)(10,34,37,62)(11,12)(13,42,45,32)(14,33)(15,22)(18,48)(20,53,26,59)(21,57,58,54)(23,55)(24,41)(27,61,56,29)(31,36,49,43)(35,64)(38,50)(39,63,44,40)(46,60)(51,52);(1,4,7,22,8,23,30,60)(2,10,13,43,14,37,45,36)(3,15,16,19)(5,26,27,57)(6,21,29,59,25,58,61,53)(9,31,32,62)(11,38,39,48)(12,18,40,50,41,51,63,64)(17,46,47,55)(20,56,54,28)(24,35,44,52)(33,49,42,34);(1,5)(2,11,14,24)(3,6,17,25)(4,20,23,26)(7,27,30,56)(8,28)(9,12)(10,35)(13,44)(15,21)(16,29)(18,49,51,31)(19,53,55,59)(22,57)(32,63,42,40)(33,41)(34,64)(36,48,43,52)(37,38)(39,45)(46,58)(47,61)(50,62)(54,60)

Polycyclic presentation:

2^11|b^2=gjk,c^2=fgk,d^2=hi,e^2=h,f^2=hjk,b^a=bg,c^a=ce,c^b=cf,d^a=dgh,d^b=dgijk,d^c=dghij,e^a=eh,e^b=ei,e^c=eh,e^d=ei,f^b=fhi,f^c=fij,f^d=fijk,f^e=fjk,g^c=gij,g^d=gjk,g^e=gjk,g^f=gj,h^b=hk,h^d=hk,i^a=ij,i^b=ijk,i^c=ijk,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286010, #72286183
maximal normal subgroup of#72286010, #72286183

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7*F.10*F.11);;
gap> SetPower(coll, 3, F.6*F.7*F.11);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
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.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);