#72226930

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	C₂	#72226926
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#491465	C₂²
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=d,d^2=hj,e^2=gi,f^2=hk,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=befhjk,c^a=cfijk,c^b=cfi,d^a=dh,d^b=dhk,e^b=eg,e^c=ek,f^a=fhjk,f^b=fhjk,f^c=fk,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226936, #72226950, #72226963
maximal normal subgroup of	#72226936, #72226950, #72226963

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