#72340555

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72210559	C₂
LCS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

5758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=hj,c^2=fk,d^2=k,e^2=j,g^2=k,c^a=ce,c^b=cf,d^a=dg,d^b=dfj,d^c=dk,e^b=eh,e^c=eh,e^d=ei,f^a=fhj,f^e=fj,g^b=ghjk,g^c=gi,g^d=gk,g^f=gk,h^c=hj,h^d=hk,i^b=ijk,i^c=ik

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340564
maximal normal subgroup of	#72340564

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