#72319858

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#106320
commutator	C₂×M₁₆	C₂×C₄×C₈
Frattini, cosocle	#33326	C₂³
socle	C₂	#72285965
prime radical	#3444	M₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ #33326 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dfijk,d^2=h,f^2=i,g^2=j,i^2=j,j^2=k,b^a=bg,c^a=ch,c^b=ce,d^b=dj,e^a=eh,e^b=ek,f^a=fj,f^b=fj,f^c=fk,f^d=fk,g^a=gk,g^b=gk,g^c=gh,g^e=gk,h^b=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319914, #72343265
maximal normal subgroup of	#72319914, #72343265

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