#72338771

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#213828
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72337659	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

5129 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:

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

Polycyclic presentation:

2^11|a^2=g,d^2=hi,f^2=jk,g^2=j,b^a=be,c^a=ch,d^a=df,d^b=dehj,d^c=dek,e^a=eh,e^d=ej,f^a=fhi,f^b=fh,f^c=fh,f^d=fjk,f^e=fjk,g^b=gh,g^c=gj,g^d=ghijk,g^e=gj,h^a=hj,h^d=hjk,i^a=ij,i^b=ik,i^c=ij,i^d=ijk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338775, #72338830
maximal normal subgroup of	#72338775, #72338830

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