Finite Group Databasebeta

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#72262534

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72262361
commutatorC2×#381C22×C4
Frattini, cosocleC2×#2613C23
prime radical#72234711C2
LCSG ▷ C2×#381 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

2964 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 2964 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^11|c^2=dfk,d^2=jk,e^2=ij,f^2=k,b^a=be,c^a=cd,c^b=ceij,d^a=djk,d^b=dg,d^c=dh,e^a=eij,e^b=eij,e^c=egijk,e^d=ei,f^a=fh,f^c=fh,f^d=fj,f^e=fijk,g^a=gi,g^c=gj,g^d=gk,g^e=gk,g^f=gk,h^b=hijk,h^c=hj,i^b=ik,i^c=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72262537, #72262552, #72262764
maximal normal subgroup of#72262537, #72262552, #72262764

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