#72286141

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

3567 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)(4,11)(5,10)(6,26,30,12)(7,13)(8,17)(9,22)(15,33,46,42)(16,47)(18,44,27,31)(19,35)(20,38)(21,37,28,41)(23,36)(24,39,29,56)(25,40)(32,48)(34,52)(43,50)(45,63,64,59)(49,58,53,57)(51,60)(54,62,55,61);(1,3,4,17)(2,8,11,14)(5,16,23,48)(6,27,28,46,30,18,21,15)(7,19,22,52)(9,35,13,34)(10,32,36,47)(12,33,41,44,26,42,37,31)(20,51,25,50)(24,49,55,45,29,53,54,64)(38,43,40,60)(39,63,62,58,56,59,61,57);(1,4)(2,9,13,11)(3,15)(5,23,25,20)(6,28)(7,22)(8,31)(10,38)(12,41,26,37)(14,42,35,33)(16,45)(17,18,52,27)(19,46)(21,30)(24,54,29,55)(32,58)(34,44)(36,40)(39,61)(43,59,47,63)(48,49,51,53)(50,64)(56,62)(57,60);(1,5,7,25)(2,10,13,40)(3,16,19,50)(4,20,22,23)(6,29)(8,32,34,60)(9,36,11,38)(12,39)(14,43,35,47)(15,45)(17,51,52,48)(18,53)(21,54)(24,30)(26,56)(27,49)(28,55)(31,57)(33,59)(37,62)(41,61)(42,63)(44,58)(46,64)

Polycyclic presentation:

2^11|a^2=g,b^2=e,c^2=ghjk,d^2=gk,e^2=g,f^2=gijk,b^a=be,c^a=chijk,c^b=cf,d^a=djk,d^b=dgj,d^c=dghijk,e^a=eg,e^c=eh,e^d=ek,f^a=fhj,f^b=fghijk,f^c=fgjk,f^d=fij,f^e=fij,g^c=gjk,g^f=gk,h^a=hk,h^b=hi,h^c=hjk,h^d=hj,h^e=hjk,h^f=hj,i^a=ij,i^b=ijk,i^c=ij,i^d=ik,i^e=ik,i^f=ik,j^b=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.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 4, F.7*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);