#72287210

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#513009	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|b^2=ij,c^2=e,d^2=efghj,e^2=g,f^2=hjk,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bij,c^a=cf,c^b=cij,d^a=dh,d^c=dh,e^a=eh,e^d=ej,f^a=fh,f^c=fj,f^d=fjk,f^e=fk,g^a=gj,g^d=gk,h^a=hjk,h^c=hk,h^d=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287443, #72323144, #72323169
maximal normal subgroup of	#72287443, #72323144, #72323169

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