#72284403

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
prime radical	#72283730	C₂
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

3519 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:

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

Polycyclic presentation:

2^11|a^2=j,b^2=gj,c^2=eghik,e^2=hik,f^2=hk,c^a=cf,c^b=ce,d^a=dg,d^b=dgjk,d^c=dgij,e^a=eg,e^b=eh,e^c=eik,e^d=eij,f^a=fh,f^b=fgik,f^c=fgh,f^d=fij,f^e=fik,g^c=gi,g^e=gj,g^f=gjk,h^b=hj,h^c=hjk,h^d=hk,h^e=hk,i^a=ij,i^b=ijk,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286327
maximal normal subgroup of	#72286327

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