#72226929

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#122887
commutator	C₈²	C₂×C₄²
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72216562
prime radical	#491465	C₂²
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

870 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=d,e^2=gi,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=befh,c^a=cij,c^b=ci,d^b=dk,e^b=eg,e^c=ek,f^a=fh,f^b=fh,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226936, #72226961, #72227242, …
maximal normal subgroup of	#72226936, #72226961, #72227242, …

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, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
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);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*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);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
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);