#72225480

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#141247
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#72216236	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

790 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=fh,d^2=gij,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cgj,c^b=cf,d^a=dg,d^b=dgj,e^b=ej,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549, #72227126, #72227142
maximal normal subgroup of	#72225549, #72227126, #72227142
maximal quotient of	#72225481

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