#72343172

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#354936
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72318993
prime radical	#3444	#41
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ #479961 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2131 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=g,d^2=hjk,e^2=i,f^2=k,h^2=j,i^2=k,j^2=k,b^a=bk,c^b=cf,d^a=dj,d^b=dijk,d^c=dh,e^d=ek,f^b=fk,f^c=fijk,f^d=fk,g^b=gij,g^f=gk,h^a=hk,h^c=hjk,j^c=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343240, #72343317, #72343482
maximal normal subgroup of	#72343240, #72343317, #72343482

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