#72251826

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72240171
commutator, Frattini, cosocle	C₂×#535	C₂⁴
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35222 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5886 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^11|b^2=k,c^2=fk,d^2=fghik,e^2=jk,i^2=k,b^a=bhk,c^a=ck,c^b=cf,d^a=dfg,d^b=dfgjk,d^c=de,e^a=eh,e^b=ei,e^c=ehik,e^d=eh,f^d=fhijk,f^e=fjk,g^b=gk,g^c=ghk,g^d=ghijk,g^e=gj,h^d=hk,i^b=ik,i^c=ij,i^d=ik,i^f=ik,i^g=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251829
maximal normal subgroup of	#72251829

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