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#72340469

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216457
commutator, Frattini, cosocle#1781C24
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

4199 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 4199 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)(3,14,17,27)(4,18)(5,11,15,10)(6,12)(7,13)(8,16,25,32)(9,20)(19,28)(21,26,30,23)(22,29)(24,31);(1,3)(2,8)(4,5,20,15)(6,16,22,32)(7,17)(9,26,18,23)(10,24)(11,28)(12,14,29,27)(13,25)(19,21)(30,31);(1,4,7,20)(2,9,13,18)(3,15,17,5)(6,19,22,31)(8,23,25,26)(10,14,11,27)(12,24,29,28)(16,30,32,21);(2,10,13,11)(8,24,25,28)(9,27,18,14)(12,23,29,26)(19,31)(21,30)

Polycyclic presentation:

2^11|a^2=i,b^2=ghijk,c^2=k,d^2=j,e^2=hj,f^2=k,b^a=bf,c^a=ck,c^b=ci,d^a=de,d^c=dhk,e^a=eh,e^b=eg,e^c=ek,e^d=ehjk,f^b=fik,f^d=fgi,f^e=fijk,g^a=gi,g^b=gjk,g^c=gk,g^d=gj,g^e=gj,h^a=hk,h^b=hj,h^f=hk,i^b=ik,i^d=ij,i^e=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340478
maximal normal subgroup of#72340478

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