#72285757

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	#72237744
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#35346	D₄
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

2095 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)(8,29)(14,41)(15,42)(16,48)(17,23)(20,27)(30,57)(31,35)(32,58)(34,39)(38,55)(43,50)(44,61)(45,62)(46,64)(47,51)(49,63)(52,53)(59,60);(1,2)(3,14)(4,18)(5,22)(6,11,28,40)(7,12)(8,16)(9,21)(10,26)(13,41)(15,45,47,44)(17,38)(19,33)(20,39,27,34)(23,55)(24,36)(25,37,56,54)(29,48)(30,46,32,49)(31,50)(35,43)(42,62,51,61)(52,60,53,59)(57,64,58,63);(1,3)(2,8,12,31)(4,15)(5,23,26,13)(6,16,28,50)(7,17)(9,32,18,30)(10,35)(11,38)(14,40)(19,51,24,42)(20,49,27,46)(21,47)(22,29)(25,43)(33,57)(34,44)(36,58)(37,41,54,55)(39,45)(48,56)(52,64)(53,63)(59,62,60,61);(1,4)(2,9,12,18)(3,15)(5,24,26,19)(6,27,28,20)(7,21)(8,30,31,32)(10,36)(11,39)(13,42,23,51)(14,44)(16,49,50,46)(17,47)(22,33)(25,52)(29,57)(34,40)(35,58)(37,59,54,60)(38,45)(41,61,55,62)(43,64)(48,63)(53,56)

Polycyclic presentation:

2^11|b^2=gjk,c^2=h,d^2=h,f^2=hik,c^a=ce,c^b=cf,d^a=dgk,d^b=dgij,d^c=dh,e^b=eg,e^c=ej,e^d=eijk,f^a=fgij,f^b=fh,f^c=fhij,f^d=fi,f^e=fij,g^c=gi,g^d=gj,g^f=gjk,h^a=hj,h^b=hjk,h^e=hk,h^f=hk,i^b=ij,i^d=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285855
maximal normal subgroup of	#72285855

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