#72225533

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#140224
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#491485	C₂²
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#7052 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

511 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:

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

Polycyclic presentation:

2^11|a^2=f,c^2=e,d^2=ghi,e^2=hik,f^2=h,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cgijk,c^b=cfgj,d^a=dg,d^b=dghj,d^c=dk,e^a=ei,e^b=ehik,f^b=fhj,g^a=gi,g^b=gi,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549
maximal normal subgroup of	#72225549
maximal quotient of	#72225534

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