#72225378

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	#72225377
prime radical	C₂³	#12047
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=d,c^2=fhik,d^2=g,e^2=gh,f^2=hk,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cfijk,c^b=cjk,d^a=dh,d^c=dk,e^b=ek,f^a=fhjk,f^b=fk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225534, #72226950
maximal normal subgroup of	#72225393, #72225534, #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.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6*F.8*F.9*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8);;
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.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
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*F.10*F.11);;
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);