#72341140

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 ◁ #20585 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3566 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=ghk,c^2=h,f^2=g,b^a=be,c^a=cf,c^b=cghij,d^a=dk,d^c=dghk,e^b=ei,e^c=eh,f^a=fg,f^b=fhj,f^c=fgik,f^d=fijk,f^e=fijk,g^b=gjk,g^c=gk,g^d=gk,g^e=gk,h^a=hi,h^b=hjk,h^d=hk,h^f=hjk,i^b=ik,i^c=ij,i^f=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341156
maximal normal subgroup of	#72341156

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