#72285162

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235715
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#422772	C₂²
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35272 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3826 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)(4,9,22,34)(5,10,24,36)(6,11,26,39)(8,27,30,17)(14,43,18,40)(15,41,23,53)(19,32,20,31)(21,38,44,33)(25,37,48,58)(28,47,57,50)(29,54,35,49)(45,61,46,62)(51,59,52,60)(55,64,56,63);(1,3,6,17)(2,8,11,13)(4,18,21,50,22,14,44,47)(5,23,25,54)(7,16,26,27)(9,28,33,40,34,57,38,43)(10,35,37,53)(12,30,39,42)(15,48,49,24)(19,46,51,64,20,45,52,63)(29,58,41,36)(31,55,59,61,32,56,60,62);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,22)(8,28,30,57)(10,31)(11,38)(12,34)(13,40,42,43)(15,45,23,46)(17,50,27,47)(20,24)(25,51)(26,44)(29,55,35,56)(32,36)(33,39)(37,60)(41,61,53,62)(48,52)(49,64,54,63)(58,59);(4,19,22,20)(9,31,34,32)(14,45,18,46)(21,51,44,52)(28,55,57,56)(33,59,38,60)(40,61,43,62)(47,63,50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 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.10*F.11);;
gap> SetPower(coll, 4, 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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
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, 3, F.5*F.11);;
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.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, 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, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);