#72226871

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₈×#357
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	C₂×#492532	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#12046 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2187 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)(3,15)(4,12)(5,26)(6,30)(7,33)(8,14)(9,18)(10,31)(11,23)(13,32)(16,46)(17,35)(19,58)(20,34)(21,44)(22,40)(24,61)(25,41)(27,38)(28,42)(29,43)(36,56)(37,52)(39,60)(45,59)(47,53)(48,55)(49,50)(51,64)(54,63)(57,62);(3,16)(4,5)(6,31)(7,32)(9,34)(10,30)(11,41)(12,26)(13,33)(15,46)(17,53)(18,20)(19,54)(21,52)(22,28)(23,25)(24,60)(27,50)(29,59)(35,47)(36,62)(37,44)(38,49)(39,61)(40,42)(43,45)(48,64)(51,55)(56,57)(58,63);(1,3,4,17,22,51,29,52,7,20,24,54,6,19,23,18,8,21,25,55,31,53,60,16,32,50,59,57,28,56,5,27)(2,9,11,36,42,63,61,49,13,37,43,47,10,35,12,44,14,38,26,58,30,62,45,34,33,46,39,64,40,48,41,15);(2,10,33,42,14,40,13,30)(9,35,46,63,38,48,37,62)(11,12,39,61,26,41,43,45)(15,47,34,36,44,64,49,58)

Polycyclic presentation:

2^11|c^2=e,d^2=gi,e^2=hjk,f^2=hk,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=cfij,d^a=dhj,d^b=dg,e^a=eh,e^b=ehk,f^a=fhjk,f^b=fhjk,f^c=fk,g^a=gjk,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226910, #72226924, #72226958
maximal normal subgroup of	#72226910, #72226924, #72226958

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