#72225241

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#513082	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|a^2=e,c^2=fh,d^2=j,e^2=g,f^2=hi,g^2=j,h^2=ik,i^2=k,c^a=chi,c^b=cf,d^b=dj,e^c=ei,f^a=fik,f^b=fh,f^e=fk,g^c=gk,h^a=hk,h^b=hi,i^b=ik

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225249, #72287443, #72323164
maximal normal subgroup of	#72225249, #72287443, #72323164

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