Finite Group Databasebeta

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#72323799

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72266314
commutator, Frattini, cosocleC2×#2613C23
prime radical#72237377C2
LCSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4548 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 4548 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:

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

Polycyclic presentation:

2^11|c^2=gi,b^a=bd,c^a=ce,c^b=cf,d^c=dg,e^b=egh,e^c=ei,e^d=ei,f^a=fhij,f^c=fjk,f^d=fj,f^e=fk,g^a=gi,g^b=gj,h^a=hk,h^b=hj,h^d=hk,i^b=ik,j^a=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72323801, #72324326, #72324333
maximal normal subgroup of#72323801, #72324326, #72324333

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