#72225385

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#2140
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225384
prime radical	C₂×#508	C₂×C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

593 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=d,c^2=e,d^2=hj,e^2=g,f^2=hk,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cfjk,c^b=ci,d^a=dh,e^a=eh,f^a=fhj,f^b=fk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72226958, #72226963
maximal normal subgroup of	#72225393, #72226958, #72226963

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