#72285984

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ #20586 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3687 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)(4,18,20,9)(5,10)(6,26,30,11)(7,12)(8,16)(14,42,27,40)(15,41,19,51)(17,43)(21,35,23,49)(22,50)(24,55,29,38)(25,37)(28,36)(31,45,39,56)(32,47,34,52)(33,48)(44,62)(46,61)(53,59)(54,60)(57,63)(58,64);(1,3,5,16)(2,8,10,13)(4,19,21,52,20,15,23,47)(6,27,29,56,30,14,24,45)(7,17,25,48)(9,34,35,41,18,32,49,51)(11,39,38,42,26,31,55,40)(12,33,37,43)(22,44,53,63)(28,46,54,64)(36,58,60,61)(50,57,59,62);(3,14)(4,20)(8,31)(9,18)(10,37)(13,40)(15,46)(16,45)(17,27)(19,44)(22,28)(32,57)(33,39)(34,58)(35,49)(36,50)(38,55)(41,61)(42,43)(47,64)(48,56)(51,62)(52,63)(59,60);(1,4)(2,9,12,18)(3,15,17,19)(5,23,25,21)(6,28,30,22)(7,20)(8,32)(10,35)(11,36)(13,41,43,51)(14,44)(16,47)(24,54)(26,50)(27,46)(29,53)(31,57,39,58)(33,34)(37,49)(38,60,55,59)(40,61)(42,62)(45,63,56,64)(48,52)

Polycyclic presentation:

2^11|a^2=g,b^2=e,d^2=gi,e^2=g,b^a=be,c^a=chi,c^b=cf,d^a=dhj,d^b=dghjk,d^c=dhij,e^a=eg,e^c=eh,e^d=eik,f^a=fh,f^b=fh,f^d=fi,f^e=fi,g^c=gjk,g^d=gk,g^f=gk,h^a=hk,h^b=hi,h^d=hj,h^e=hjk,i^a=ij,i^b=ijk,i^d=ik,i^e=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311013, #72340399, #72340462, …
maximal normal subgroup of	#72311013, #72340399, #72340462, …

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