Finite Group Databasebeta

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#72227531

order2048
nilpotency class7
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227525
commutator, Frattini, cosocle#8541C23
prime radicalC2×#2846D4
LCSG ▷ #8541 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C25 ◁ C23×D4 ◁ #8541 ◁ G
LECSG ▷ #8541 ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2509 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 2509 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=fhijk,b^2=gij,c^2=gi,d^2=fijk,e^2=j,g^2=k,b^a=be,c^a=cd,c^b=cgijk,d^a=df,d^b=dfgk,d^c=dfhij,e^a=eij,e^b=eh,e^c=efgij,e^d=ehijk,f^a=fjk,f^b=fij,f^c=fj,f^d=fk,f^e=fj,g^a=gh,g^b=gk,g^d=gij,g^e=gjk,g^f=gk,h^a=hk,h^b=hjk,h^c=hi,h^d=hjk,i^a=ij,i^b=ik,i^d=ik,j^c=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227538, #72227548

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