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#72268792

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#93139
commutatorC23×C4⋊C4C42
Frattini, cosocleC2×#15155C22
prime radicalC24×D4C22⋊C4
LCSG ▷ C23×C4⋊C4 ▷ C24×C4 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4 ◁ C2×#15155 ◁ G
LECSG ▷ C2×#15155 ▷ C24×C4 ▷ C24 ▷ C2 ▷ C1

2475 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 2475 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=e,c^2=hij,d^2=hk,f^2=k,b^a=bcdhk,c^b=cf,d^a=dfhij,d^b=dfg,d^c=dhk,e^a=eghij,e^c=eik,e^d=eijk,f^a=fh,f^b=fi,f^c=fk,f^e=fk,g^a=gik,g^b=gj,h^a=hk,h^b=hk,i^b=ik,j^a=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268798
maximal normal subgroup of#72268798

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