#72340949

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#213828
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#72337659	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂⁴ ▷ C₂² ▷ C₁

4713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=k,e^2=k,f^2=j,b^a=bf,c^a=cg,c^b=cjk,d^a=dgjk,d^b=dgk,e^b=eh,e^c=ei,e^d=ei,f^a=fhj,f^b=fj,f^c=fj,f^d=fi,g^a=gi,g^e=gk,g^f=gj,h^c=hj,h^d=hjk,i^a=ik,i^b=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340954
maximal normal subgroup of	#72340954

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