#72340471

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20586 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3681 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)(3,14,15,41)(4,19)(5,11,26,9)(6,12)(7,13)(8,18,34,27)(10,24,39,21)(16,46,20,54)(17,43)(22,53)(23,35)(25,40,29,36)(28,58)(30,42)(31,57)(32,59)(33,50)(37,63)(38,56,64,55)(44,47)(45,62)(48,52,51,60)(49,61);(1,3)(2,8,13,34)(4,20,23,16)(5,17,26,45)(6,27,30,18)(7,15)(9,33,11,31)(10,37)(12,41)(14,42)(19,52)(21,47)(22,48)(24,49)(25,50)(28,51)(29,57)(32,39)(35,60)(36,62)(38,44,64,61)(40,43)(46,58,54,53)(55,59,56,63);(2,9,13,11)(3,15)(8,31,34,33)(10,19,39,35)(12,40,42,36)(14,43,41,62)(16,20)(17,45)(18,27)(22,28)(25,29)(32,52,37,60)(38,53,64,58)(44,54,61,46)(47,49)(59,63);(1,4,7,23)(2,10,13,39)(3,16,15,20)(5,24,26,21)(6,28)(8,32)(9,35,11,19)(12,38)(14,44,41,61)(17,49,45,47)(18,51)(22,30)(25,56)(27,48)(29,55)(31,52)(33,60)(34,37)(36,58)(40,53)(42,64)(43,46,62,54)(50,63)(57,59)

Polycyclic presentation:

2^11|a^2=ik,b^2=ghijk,c^2=j,d^2=hj,e^2=hj,f^2=k,b^a=bf,c^a=ce,c^b=ck,d^a=dek,d^b=dijk,d^c=dhk,e^a=eh,e^b=eg,e^c=ehjk,e^d=ehj,f^b=fik,f^c=fgi,f^d=fgi,f^e=fijk,g^a=gi,g^b=gjk,g^c=gj,g^d=gjk,g^e=gj,h^a=hk,h^b=hj,h^f=hk,i^b=ik,i^c=ij,i^d=ij,i^e=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340478
maximal normal subgroup of	#72340478

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