#72285086

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235719
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#417566	C₂²
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35272 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=g,c^2=i,e^2=k,g^2=i,b^a=bg,c^a=ci,d^a=df,d^b=dfj,d^c=de,e^a=ei,e^b=eik,e^d=ek,f^b=fh,f^c=fi,f^e=fk,g^a=gi,g^d=ghk,g^f=gjk,h^a=hj,h^b=hjk,h^g=hk,i^d=ik,j^b=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286380
maximal normal subgroup of	#72286380

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