#72225372

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#2177
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂⁴
socle	C₂	C₂×#122946
prime radical	#33981	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=fhk,d^2=g,e^2=g,f^2=hk,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cfjk,c^b=cjk,d^a=dhk,e^a=eh,e^c=ek,f^a=fhj,f^b=fk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225515, #72226924
maximal normal subgroup of	#72225393, #72225515, #72226924

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