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#72285890

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237744
commutator, Frattini, cosocle#1781C24
prime radical#72241387C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2595 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 2595 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8,15,10)(4,16)(5,22)(6,24)(7,11)(9,19)(12,27)(13,14,23,28)(17,31,30,18)(20,26,32,29)(21,25);(1,3)(2,8,11,10)(4,17)(5,23,24,13)(6,14)(7,15)(9,26,27,29)(12,20,19,32)(16,18)(21,30)(22,28)(25,31);(1,4,6,20)(2,9,24,31,11,27,5,18)(3,12,14,30,15,19,28,17)(7,21,22,32)(8,25,23,26)(10,16,13,29);(1,5,7,24)(2,6)(3,13,15,23)(4,18,21,31)(8,14)(9,20)(10,28)(11,22)(12,26,19,29)(16,30)(17,25)(27,32)

Polycyclic presentation:

2^11|a^2=gi,b^2=ghik,c^2=fghijk,d^2=hi,e^2=hijk,f^2=hjk,b^a=bghk,c^a=ce,c^b=cf,d^a=dij,d^b=dghk,d^c=dg,e^a=eh,e^b=eg,e^c=eghij,e^d=ejk,f^a=fg,f^b=fhi,f^c=fij,f^d=fijk,f^e=fi,g^a=gj,g^b=gj,g^c=gi,g^d=gjk,g^e=gj,g^f=gjk,h^a=hjk,h^b=hk,h^d=hk,h^e=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

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286183, #72338995
maximal normal subgroup of#72286183, #72338995

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, 1, F.7*F.9);;
gap> SetPower(coll, 2, F.7*F.8*F.9*F.11);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.11);;
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.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
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.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, 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);