#72284062

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

Polycyclic presentation:

2^11|a^2=hj,b^2=gh,c^2=i,d^2=hk,e^2=hk,b^a=bhik,c^a=ce,c^b=cf,d^b=dgik,d^c=dhj,e^a=eh,e^b=eg,e^c=ehjk,f^a=fgijk,f^b=fi,f^c=fk,f^d=fijk,f^e=fijk,g^a=gjk,g^b=gj,g^c=gi,g^d=gjk,g^e=gj,h^b=hj,h^f=hk,i^a=ij,i^b=ik,i^d=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285484
maximal normal subgroup of	#72285484

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