#72227036

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227035
commutator	C₁₆²	C₂×C₄
Frattini, cosocle	C₁₆×C₃₂	C₂²
prime radical	#72216263	C₂
LCS	G ▷ C₁₆² ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×C₁₆ ◁ C₁₆×C₃₂ ◁ G
LECS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,c^2=efg,d^2=f,e^2=gi,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bc,c^a=ce,c^b=cefh,d^b=dfh,e^a=eg,e^b=eg,f^b=fh,g^a=gi,g^b=gi,h^b=hj,i^a=ik,i^b=ik,j^b=jk

Conjugacy classes & character table

179 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227042, #72227076, #72227080
maximal normal subgroup of	#72227042, #72227076, #72227080

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