#72234066

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#363895
commutator	C₂²×C₄⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
prime radical	#72233824	C₂
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 6233 conjugacy classes of subgroups; only the 392 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^11|a^2=h,b^2=f,c^2=efhi,e^2=k,g^2=j,b^a=bg,c^a=cg,c^b=ce,d^a=dh,d^c=dk,e^a=ei,e^b=ei,e^c=eij,f^c=fik,f^e=fk,g^b=gj,g^c=gij,g^d=gj,g^e=gk,h^b=hj,h^c=hj,i^b=ik,i^c=ik

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234105
maximal normal subgroup of	#72234105

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