#72225387

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	C₈	C₄×#357
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225386
prime radical	C₂×#508	C₂×C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₆₄ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

581 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=d,c^2=e,d^2=hjk,e^2=g,f^2=hk,g^2=i,h^2=jk,i^2=k,j^2=k,b^a=bf,c^a=ch,c^b=ci,d^a=dh,d^c=dk,e^a=ejk,e^b=ek,f^a=fhjk,f^b=fk,g^a=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72226924, #72226950
maximal normal subgroup of	#72225393, #72226924, #72226950

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