#72225583

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#123363
commutator, Frattini, cosocle	C₈×C₁₆	C₂⁴
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=fhk,e^2=gi,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=befh,c^a=cfk,c^b=cfk,d^a=dk,d^c=dk,e^b=eg,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225593, #72226936, #72226950
maximal normal subgroup of	#72225593, #72226936, #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, 3, F.6*F.8*F.11);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
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.5*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*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.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);