#72343458

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285170
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#151182	C₄
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #479958 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,5,12,7,14,27,36,10,3,8,17,9,15,28,25)(4,16,20,49,22,48,23,43,24,11,37,40,38,42,61,50)(6,29,33,62,34,18,51,44,31,47,26,63,35,13,45,30)(19,53,54,39,55,56,52,64,41,32,60,21,57,58,46,59);(1,3)(2,10)(4,19,24,41)(5,25)(6,30)(7,14)(8,36)(9,15)(11,39,16,21)(12,28)(13,26)(17,27)(18,33)(20,46,37,52)(22,57,38,55)(23,60,61,54)(29,51)(31,44)(32,40,53,49)(34,63)(35,62)(42,59,48,64)(43,56,50,58)(45,47);(6,31)(13,18)(26,33)(29,47)(30,44)(34,35)(45,51)(62,63);(1,4)(2,11)(3,16)(5,20)(6,32,31,53)(7,22)(8,37)(9,38)(10,24)(12,40)(13,46,18,52)(14,42)(15,48)(17,49)(19,44,41,30)(21,26,39,33)(23,27)(25,43)(28,61)(29,54,47,60)(34,58,35,56)(36,50)(45,64,51,59)(55,63,57,62)

Polycyclic presentation:

2^11|a^2=e,b^2=hj,d^2=g,e^2=h,f^2=ijk,h^2=k,i^2=jk,j^2=k,b^a=bik,d^a=dk,d^b=df,d^c=dg,e^b=ejk,f^a=fjk,f^b=fi,f^c=fhjk,f^d=fhijk,f^e=fk,g^b=ghj,g^f=gk,h^b=hk,i^a=ik,i^b=ij,i^d=ij,j^b=jk,j^d=jk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343465
maximal normal subgroup of	#72343465

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