#72226927

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226926
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂×#12515	C₂²
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=d,d^2=hjk,e^2=gi,f^2=h,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=befhj,c^a=cfi,c^b=cfij,d^a=dh,d^b=dhk,e^b=eg,e^c=ek,f^a=fhj,f^b=fhj,f^c=fk,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226936, #72226950, #72226958
maximal normal subgroup of	#72226936, #72226950, #72226958

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