#72227000

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×#357
commutator	C₂×C₁₆	C₂²×C₁₆
Frattini, cosocle	C₁₆²	C₂³
socle	C₂²	C₂×#7068
prime radical	C₂²×D₁₆	C₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ C₂×C₈×C₁₆ ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=d,c^2=hj,d^2=f,e^2=gk,f^2=h,g^2=i,h^2=j,i^2=k,b^a=be,c^a=cik,c^b=cj,d^a=dg,e^a=egi,e^b=ek,f^a=fi,g^a=gik,h^a=hk,i^a=ik

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227028, #72227114, #72227126
maximal normal subgroup of	#72227028, #72227114, #72227126
maximal quotient of	#72227001

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