#72225376

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#12674
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225375
prime radical	C₂³	#12011
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

475 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=d,c^2=fh,e^2=g,f^2=hj,g^2=i,h^2=jk,i^2=k,j^2=k,b^a=bi,c^a=cf,c^b=cjk,d^a=dk,d^c=dk,e^b=ek,f^a=fh,f^b=fk,h^a=hj,j^a=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225523, #72226936, …
maximal normal subgroup of	#72225393, #72225523, #72226936, …

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