#72225514

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#138625
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#72225429	C₂
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

747 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)(36,49)(37,52)(38,44)(40,51)(41,45)(42,50)(43,47)(46,55)(48,57)(53,61)(54,62)(56,64)(58,60)(59,63);(1,2)(3,12)(4,16)(5,19)(6,11)(7,13)(8,17)(9,24)(10,20)(14,30)(15,21)(18,25)(22,32)(23,29)(26,28)(27,31)(34,39)(35,45)(36,47)(37,50)(40,56)(41,43)(42,57)(44,55)(46,60)(48,64)(49,54)(51,63)(52,53)(58,59)(61,62);(1,3,4,13,5,14,17,28,6,15,18,29,20,31,24,32)(2,7,8,21,10,22,16,30,11,23,9,12,19,26,25,27)(34,40,42,43,44,49,52,59)(35,46)(39,41,53,56,55,58,57,54)(45,60)(48,61)(62,64)

Polycyclic presentation:

2^11|c^2=e,d^2=gi,e^2=h,f^2=h,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cgk,c^b=cf,d^a=dg,d^b=dg,d^c=dk,e^a=eik,e^b=eh,f^a=fik,f^b=fhj,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549
maximal normal subgroup of	#72225549
maximal quotient of	#72225515

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, 3, F.5);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8);;
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.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*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, 1, F.8*F.11);;
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);